Edge covering pseudo-outerplanar graphs with forests

Edge Co v ering Pseudo-outerplanar Graphs with F orests ∗ Xin Zhang, Guizhen Liu † and Jian-Liang W u School of Mathematics, Shandong Universit y , Ji nan, 250100, China Abstract A graph is called pseudo-outerplanar if eac h block has an embedding on t h e plane in such a wa y that the vertice s lie on a fixed circle and the edges lie insi de the disk of this circle with ea c h of them crossing at most one another. In this pap er, we prov e that each pseudo-outerplanar graph admits edge decomp ositions into a linear forest and an outerplanar graph, or a star forest and an outerplanar graph, or tw o forests and a matching, or max { ∆( G ) , 4 } matc hings, or max {⌈ ∆( G ) / 2 ⌉ , 3 } linear forests. These results generali ze some ones on outerplanar graphs and K 2 , 3 -minor-free graphs, since t he class of pseudo-outerplanar graphs is a larger class than the one of K 2 , 3 -minor-free graphs. Keywor ds : pseudo-outerplanar graphs; edge d ecomp osition; edge chromatic n u m b er; linear arboricit y . 1 In tro duction In this pap er, all graphs considered are finite, simple and undirected. W e use V ( G ) , E ( G ) , δ ( G ) and ∆( G ) to denote the vertex set, the edge set, the minim um degree and the ma xim um degr ee of a graph G , resp ectively . Let d G ( v ) (or d ( v ) for simplicit y) denote the de gr e e of a v ertex v ∈ V ( G ) . A bl ock is a maximal 2 -connected subgraph of a given gra ph G . A gr aph H is a m i nor of a gra ph G if a copy of H can b e obtained fro m G v ia rep eated edge deletion and/or edge contraction. F o r a subset S ⊆ V ( G ) ∪ E ( G ) , G [ S ] denotes the subgraph of G induced by S . The vertex c onne ctivity of a gra ph G , denoted by κ ( G ) , is the minimum n umber o f vertices whose deletion from G disconnects it. F or other undefined concepts w e refer the reader s to [3]. An outerplanar gr aph is a gr aph that can be embedded on the plane in such a w ay that it has no cr ossings and that all its vertices lie on the outer face. In this pap er, we aim to introduce an extensio n of this concept. A graph is called pseudo-outerplanar if each block has an e mbedding on the plane in s uc h a w ay that the vertices lie on a fixe d circle and the edges lie inside the disk of this circl e with each of them crossing at most one another. In this embedding, the edges b ounding the disk(s) are ca lled b oundary e dges and a disk is sa id to be close d or op en according to whether or not it contains the circle that constitutes its b oundar y . F or exa mple, Figure 1 exhibits a pseudo-outerplana r em b edding o f a gr aph with tw o blocks: one is K 4 and the o ther is K 2 , 3 . The drawing of K 4 in this embedding lies inside a c lo sed disk but the one o f K 2 , 3 in this embedding lies inside an op en disk. In Figure 1, the edges in b old are the bo undary edges. A pseudo-outerpla nar graph is maximal if it is not p oss ible to a dd an edge suc h that the resulting graph is still pseudo- outerplanar . Thus K 2 , 3 is no t a maxima l pseudo- o uterplanar graph, since we can p os s ibly a dd tw o edges to K 2 , 3 and rema in its pseudo-outerplanarity . One ca n easily chec k that each pseudo-outer pla nar graph has a planar embedding by its definition. So the cla ss of pse udo -outerplanar graphs forms a subclas s of planar gra phs. Actually , the definition of pseudo-outerplanar graphs are simila r to that of 1-plana r g raphs (i.e. gr aphs that can b e dr awn on the pla ne so that each edge is crossed by at most one o ther edge), whic h was intro duced by Ringel [10]. Many classic problems in graph theory are considered for the cla ss of planar g r aphs and its sub classes, s uc h as the class of ser ies-para llel graphs and the o ne of outerplanar gra phs. T aking the problem o f cov er ing graphs with E-mails: sdu.zhang@y ahoo.com.cn (X. Zhang), gzliu@sdu.edu.cn (G. Liu), jlwu@sdu.edu.cn (J. L. W u). ∗ Researc h supported by NSF C (10971121, 11101243, 61070230), RFDP (2010013 1120017 ) and GI IFSDU (yzc10040). † Corresponding author. 1 Figure 1: An example of pseudo-outerplanar forests and a graph of b ounded maximum degree for example, we say that a gr a ph is ( t, d ) - cov e rab l e if its edges can be cov er ed by at most t fore s ts and a graph of maximum degree d . In [2], Ba logh et al. co njectured that ev ery simple planar graph is (2 , 4) -coverable a nd gav e a example to sho w that there are infinitely man y plana r graphs that ar e not (2 , 3) -cov er able. This conjecture was recently confirmed by Gonçalves in [5]. In [2], it is als o prov ed that ev ery s e ries-para llel gr aph is (2 , 0 ) -cov era ble a nd that every K 2 , 3 -minor-free graph is b oth (1 , 3) -cov erable and (2 , 0) -coverable. Since a gr aph is outer planar if a nd only if it is { K 4 , K 2 , 3 } -minor-free [8], ev ery outerpla nar graph is b oth (1 , 3 ) -cov era ble and (2 , 0) -cov erable. It is interesting to know what can b e said ab out pseudo-o uterplanar graphs, a nother larger cla ss than outerplanar graphs. Edge-coloring is a no ther classic pro blem in gra ph theory . In fact, we can r e gard edge-colo ring pro blems as a cov ering problem. When we co lor the edges of a g raph G , o ur a ctual task is to decompo se the edge set E ( G ) into some parts suc h that the g raph induced by e a ch part satisfies a pr op e rty P . Differen t proper ties P co rresp ond to different t ype s of edge-c o loring. F o r example, a pr op er k - e dge-c oloring o f G is a decomp osition o f E ( G ) into k subsets such that the graph induced by each subset is a match ing in G . The minim um integer k such that G has a prop er k -edge- coloring, denoted by χ ′ ( G ) , is the e dge chr omatic numb er of G . Vizing’s Theor em states that fo r any graph G , ∆( G ) ≤ χ ′ ( G ) ≤ ∆( G ) + 1 . A graph G is said to b e of class 1 if χ ′ ( G ) = ∆( G ) , and of class 2 if χ ′ ( G ) = ∆( G ) + 1 . T o determine whether a planar g raph is of cla ss 1 is a n in tere sting problem. Sa nder s and Zhao [11] show ed that each planar g raph with maximum degree at lea st 7 is o f class 1. Juv an, Mohar and Thomas [9] prov ed that ea ch series-para llel gr aph with maximum degree at least 3 is of class 1, and th us holds for o uter planar graphs. It is o p en whether each pseudo-outerplana r graph wi th la rge max imum deg ree is of class 1. On the other hand, one can consider improp er edge-colo rings. Concerning this topic, Harary [7] in tro duced the concept of linea r a rb oricity . A line ar for est is a fo r est in which every co nnected comp onent is a path. A k -tr e e- c oloring of G is a decomp osition of E ( G ) in to k subsets such that the graph induced b y each subset is a linear forest. The line ar arb oricity la ( G ) of a g raph G is the minimu m integer k suc h that G has a k -tree-co loring. Akiyama, Exo o and Harary [1] co njectured that l a ( G ) = ⌈ (∆( G ) + 1) / 2 ⌉ for a ny regular g raph G . It is obvious that l a ( G ) ≥ ⌈ ∆( G ) / 2 ⌉ for any graph G and l a ( G ) ≥ ⌈ (∆( G ) + 1) / 2 ⌉ for any regular graph G . Hence the conjecture is equiv alent to the following one. Conjecture 1.1 (Linear Arbo r icit y Co njecture) . F or any gr aph G , ⌈ ∆( G ) 2 ⌉ ≤ la ( G ) ≤ ⌈ ∆( G )+1 2 ⌉ . Now Conjecture 1.1 has b een prov ed true for all planar g raphs (see [13, 15]). How ever, it is still interesting to determine whic h kinds of planar gr aphs satisfy l a ( G ) = ⌈ ∆( G ) / 2 ⌉ . W u [13] prov ed tha t it holds for planar graphs with maxim um degree at lea s t 13. And the b ound 13 w as later impr ov ed to 9 by Cygan et al. [4]. F o r s ub clas ses o f planar graphs, W u [14] prov ed that la ( G ) = ⌈ ∆( G ) / 2 ⌉ for all series- parallel gra phs (hence also for all outerplana r graphs) with maximum degr ee a t le a st 3. Can the same co nclusion extend to the class of ps eudo-outerplanar graphs? In Section 2, w e g ive some r elationships among three clas ses containing the outerplanar gra phs; they are the K 2 , 3 -minor-free graphs, the series-pa rallel graphs and the pse udo-outerplanar g raphs. In Section 3 , we in vestigate the problem o f covering ps e udo -outerplanar g r aphs with fores ts and a graph o f b ounded maximum degr ee. In Section 4, some unav oida ble structur es o f pseudo-o uterplanar graphs are obtained. Thes e structures will be applied 2 Þ v 1 v 2 v i - 1 v i v n v n - 1 v k + 2 v k + 1 v j + 1 v j + 2 v k - 1 v k v j v j - 1 v i + 2 v i + 1 v j v j - 1 v i + 2 v i + 1 v 1 v 2 v i - 1 v i v j + 1 v j + 2 v k - 1 v n v n - 1 v k + 2 v k + 1 v k Figure 2: Each hamiltonian pseudo- outerplanar gr aphs has a hamiltonian diagram to determine the edge ch romatic num b er and linear arb oricity of pseudo- outerplanar gr aphs in Section 5. 2 Basic Prop e rties Let G b e a pseudo-o uterplanar graph. In the following of this pa p er , we alw ays ass ume that G ha s b een drawn on the plane such that (1) for each block B of G , the vertices of B lie on a fixe d circle and the edges of B lie ins ide the disk o f this circle with eac h of them crossing at most one a no ther; (2) the n umber of cro ssings in G is as small as p ossible. This drawing is called a pseudo - ou terpl an ar diag r am of G . Let G be a pseudo-outerplana r diagr am and let B b e a blo ck of G . Denote by v 1 , v 2 , · · · , v | B | the vertices of B , which ar e lying in a clo ckwise s equence. Let V [ v i , v j ] = { v i , v i +1 , · · · , v j } and V ( v i , v j ) = V [ v i , v j ] \{ v i , v j } , where the subscr ipts and the additions are tak en mo dular | B | . Lemma 2.1. [8] L et G b e an outerplanar gr aph. Then (a) δ ( G ) ≤ 2 , (b) κ ( G ) ≤ 2 . Theorem 2.2. L et G b e a pseudo-outerplanar gr aph. Then (a) δ ( G ) ≤ 3 , (b) κ ( G ) ≤ 2 u nless G ≃ K 4 . Pr o of. The pro o f of (a) is left to Coro llary 4.3. So we only prov e (b) here. If | G | ≤ 4 , then this theo r em is trivia l. So we as sume that G is a pseudo-outerplanar diagram with | G | ≥ 5 a nd κ ( G ) ≥ 3 . If G has no cross ing s, then G is an o uterplanar gra ph and th us by Lemma 2.1, κ ( G ) ≤ 2 , a con tradiction. So w e as sume that ther e are t wo chords v i v j and v k v l in G that cross each o ther, and that v i , v k , v j , v l are lying in a clo ckwise s equence. Since | G | ≥ 5 , at least one o f V ( v i , v k ) , V ( v k , v j ) , V ( v j , v l ) and V ( v l , v i ) is nonempty . Without loss of g enerality , assume that V ( v i , v k ) 6 = ∅ . Since v i v j crosses v k v l , there is no edges b etw een the t wo vertex sets V ( v i , v k ) a nd V ( v k , v i ) . So { v i , v k } s eparates V ( v i , v k ) and V ( v k , v i ) , con tradicting to κ ( G ) ≥ 3 . It is well-kno wn that ev ery 2 -connected outerplanar graph is hamiltonian. But this result do es not hold for 2 - connected pseudo -outerplanar gr aphs. The complete bipartite gr a ph K 2 , 3 is such a coun ter example. A 2 -c o nnected pseudo-outerplana r diagr am is ca lled a hamil tonia n diag ram if it is in s uch a wa y that a ll its v er tices lie o n a clo se d cir cuit C (i.e. the disk o f C is closed). This closed circuit C is called the hamil tonian boundar y of the diagra m. By this definition, one can ea sily see that a non-hamiltonian 2 -connected pseudo - outerplanar graph cannot hav e a hamiltonian diagr am. It seems interesting to answer whether ea ch hamiltonian pseudo-o uter planar graph has a hamiltonian dia gram. Theorem 2.3. L et G b e a pseudo-outerplanar diagr am and C b e a hamiltonian cycle of G . If C is not the b oundary of G , then G has a hamiltonian diagr am such that C is the hamiltonian b oundary of this diagr am. 3 Pr o of. W e pro ceed b y induction on the order of G . Since G has a hamiltonian cycle C = v 1 v 2 · · · v n v 1 that is not the b o undary of the pseudo-outerplanar diagram o f G , one can easily deduce that there exis ts at least o ne crossing in the dra wing of C (a sub-diagram of G indeed). Supp ose that v j v j +1 and v k v k +1 (j 2 n − 2 . Hence, the g raph G n ( n ≥ 6 ) cannot be covered b y tw o forests. F rom Corollary 3 .6 and Theorem 3.7, we direc tly hav e the following tw o coro llaries. Corollary 3.8. Every pseudo-outerplanar gr aph is (2 , 1) -c over able; the two p ar ameters given her e ar e b est p ossible. Corollary 3. 9. The arb oricity of a pseudo-outerplanar gr aph is at most 3 ; and this b ound is sharp. 4 Una v oidabl e Structur es In this section, a vertex set V [ v i , v j ] ( i < j ) is called a non- e dge if j = i + 1 and v i v j 6∈ E ( G ) , called a p ath if v k v k +1 ∈ E ( G ) for all i ≤ k < j and called a subp ath if j > i + 1 and some edges in the for m v k v k +1 ( i ≤ k < j ) are missing . W e say a chord v k v l ( k < l ) is contained in a ch ord v i v j ( i < j ) if i ≤ k and l ≤ j . In any figure of this section, the solid vertices hav e no edges of G inciden t with them other than those shown. Lemma 4.1. [12] L et G b e a 2 -c onn e cte d outerplanar gr aph. Then (1) G has two adjac ent 2 -vertic es u and v , or (2) G has a 3 -cycle uw xu such t hat d ( u ) = 2 and d ( w ) = 3 , or (3) G has a 4 -vertex w , wher e N ( w ) = { u, v , x, y } , such that d ( u ) = d ( v ) = 2 , N ( u ) = { w, x } and N ( v ) = { w, y } . F or the c la ss of pseudo-o uterplanar g raphs, we have a similar structural theorem as Lemma 4.1. But it seems m uch more co mplex since cro ssings are p ermitted in a pseudo-o uterplanar gr a ph. Theorem 4 .2. L et G b e a pseudo-outerplanar diagr am with δ ( G ) ≥ 2 . Then G c ontains one of the fol lowing c onfigur ations G 1 – G 17 . Mor e over, (a) if G c ontains some c onfigur ation among G 6 – G 17 , then the dr awing of this c onfi gur ation in the figur e is a p art of the diagr am of G with its b ending e dges c orr esp onding t o the chor ds; (b) if G c ontains t he c onfigur ation G 3 and xy 6∈ E ( G ) , wher e x and y ar e t he vertic es of G 3 as describ e d in the figur e, then we c an pr op erly add an e dge xy to G so that the r esult ing diagr am is stil l pseudo-outerplanar. u v u v x y u v x y u v x y w u v w x y z u v 0 x 0 y u v w x y u v w x y u v w x y z u v w x y z u v w x y z a u v w x y x u v y w u v w u v w u v w x y z u v w x y z a 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8 G 9 G 10 G 11 G 12 G 13 G 14 G 15 G 16 G 17 G 8 Pr o of. W e first co ns ider the case when G is a 2 - connected pseudo-o uterplanar diagram. Reca ll that this diagr am minimizes the n umber of c r ossings. Let v 1 , v 2 , · · · , v | G | be the vertices of this diagram lying in a clo ckwise sequence. If there is no crossing s in G , then G is an outerpla nar graph and th us G s a tisfies this theorem by Lemma 4.1. Otherwise, w e can prop erly choose o ne chord v i v j such that (1) v i v j crosses v k v l in G ; (2) v i , v k , v j and v l are lying in a clo ckwise sequence; (3) b esides v i v j and v k v l , there is no crossed c ho rds in C [ v i , v l ] . The condition (3) can b e eas ily fulfilled, b e c a use otherwise we could c hange the v alues o f i and j to meet this condition (note that the v alues of k and l are determined by i and j ). Without loss of g enerality , assume that 1 ≤ i < k < j < l ≤ | G | , b ecause otherwise w e ca n adjust the lab ellings o f the vertices in G to meet it. Claim 1 . V [ v i , v k ] is either non-e dge or p ath, and so do V [ v k , v j ] and V [ v j , v l ] . W e only need to prove that V [ v i , v k ] cannot b e subpath. Otherwise there exists tw o vertices v m and v m +1 , where i ≤ m ≤ k − 1 , s uc h that v m v m +1 6∈ E ( G ) . If there ar e chords in the form v a v m +1 such that i ≤ a ≤ m − 1 , then we choose one among them such that a is maximum. O ne can see that v a is a vertex cut of G , b ecause there is no edg es b etw een V [ v a +1 , v m ] and V [ v m +1 , v a − 1 ] by the ch oice of a a nd (3). This contradicts the fact that G is 2 -connected. Thus there is no chords in the form v a v m +1 such that i ≤ a ≤ m − 1 . Similarly , there is no c ho rds in the form v m v b such that m + 2 ≤ b ≤ k . Let p = max { n | v m +1 v n ∈ E ( G ) , m + 1 < n ≤ k } a nd q = min { n | v n v m ∈ E ( G ) , i ≤ n < m } . Since V [ v i , v k ] is neither non-edge nor path, w e hav e k − i ≥ 2 and th us at least one of the integers p a nd q exists. Without loss of ge ner ality s uppos e that p exists. Then v p is a vertex cut of G , b e cause there is no edges betw een V [ v m +1 , v p − 1 ] and V [ v p +1 , v m ] by the choices of m , p a nd by (3). This contradiction completes the pro of of Claim 1 . Claim 2 . If V [ v i , v k ] is a p ath and k − i ≥ 3 , t hen G has a sub gr aph isomorph ic to one of the c onfigur ations { G 1 , G 2 , G 4 } . This r esult also holds for V [ v k , v j ] and V [ v j , v l ] if j − k ≥ 3 and l − j ≥ 3 , r esp e ctively. Suppos e that ther e is no other chord except v i v k (if exists) in V [ v i , v k ] , then the configur ation G 1 o ccurs, since k − i ≥ 3 . So we assume that S := C [ v i , v k ] \ { v i v k } 6 = ∅ . Now w e pr ov e that there ex ists at least one chord in S that contains at least one other c hord. Supp ose that suc h a chord does not exist. Then w e first choose a chord v m v n ∈ S ( m < n ) . Without lo ss o f generality , assume that n 6 = k . If n − m ≥ 3 , then we can ea sily find a copy of G 1 in G , since v m v n contains no other ch ords b y our assumption. If n − m = 2 , then it is trivia l to see that d ( v m +1 ) = 2 . Now if min { d ( v m ) , d ( v n ) } ≤ 3 , then a copy o f G 2 would be found. Thus w e shall assume that min { d ( v m ) , d ( v n ) } ≥ 4 . So there exists a nother chord v n v p ( n < p ) in S , since d ( v n ) ≥ 4 a nd v m v n cannot be cont ained in a chord in the form v q v n ( q < n ) b y the a ssumption. Similarly , we shall assume that p − n = 2 and d ( v n +1 ) = 2 for otherwise the co nfiguration G 1 would b e fo und. Now one can see that d ( v n ) = 4 , beca use otherwise there would be chord in S that con ta ins either v m v n or v n v p , a c o nt radiction. Therefor e, the graph induced b y V [ v m , v p ] con tains the configuration G 4 . Thus we can c ho o se one chord v a v b ∈ S ( a < b ) such that v a v b contains at least one chord, and furthermore, ev ery chord co nt ained in v a v b contains no other chords (this condition can b e easily fulfilled by prop erly changing the v alues of a and b if necessary ). Let v m v n ( m < n ) be the chord cont ained in v a v b . Then by the similar argument as ab ov e, we hav e to consider the case when n − m = 2 , d ( v m +1 ) = 2 and min { d ( v m ) , d ( v n ) } ≥ 4 . Without loss of generality , a ssume that n 6 = b . Then there mu st b e a chord v n v p ( n < p ≤ b ) in S , since d ( v n ) ≥ 4 and v m v n can not b e co nt ained in a chord in the form v q v n ( q < n ) b y the choices of a and b . By the similar argument as befor e, if G co nt ains no co pies o f G 1 or G 2 , then p − n = 2 and d ( v n +1 ) = 2 . F urthermore , one can similar ly prov e that d ( v n ) = 4 by the choices of a and b . Thu s we w ould find a copy of G 4 in the g r aph induced by V [ v m , v p ] . Claim 3 . A t most one of V [ v i , v k ] , V [ v k , v j ] and V [ v j , v l ] c an b e non- e dge. If V [ v i , v k ] and V [ v k , v j ] are non-edg e , then it is trivial that v l is a vertex cut of G , contradicting the fact that G is 2 -connected. If V [ v i , v k ] and V [ v j , v l ] are non-edg e , then we c a n adjust the dr awing of G b y replacing the vertices order { v i , v k , v k +1 , · · · , v j − 1 , v j , v l } with { v i , v j , v j − 1 , · · · , v k +1 , v k , v l } . This opera tion can reduce the n um b er of 9 crossings in the drawing of G b y one, contradicting the assumption that this diagr a m minimizes the n umber of crossings . Claim 4 . If one of V [ v i , v k ] , V [ v k , v j ] and V [ v j , v l ] is non-e dge, then G has a sub gr aph isomo rphic to one of the c onfigur ations { G 1 , G 2 , G 3 } . Suppos e that V [ v i , v k ] is a non-e dg e. By Claims 1– 3, b oth V [ v k , v j ] and V [ v j , v l ] are paths with 1 ≤ j − k ≤ 2 and 1 ≤ l − j ≤ 2 . If j − k = 2 and v k v j ∈ E ( G ) , then it is clea r that d ( v k ) = 3 and d ( v k +1 ) = 2 , implying that the configuration G 2 o ccurs. If j − k = 2 but v k v j 6∈ E ( G ) , then d ( v k ) = d ( v k +1 ) = 2 , implying that the co nfiguration G 1 o ccurs. So we a ssume that j = k + 1 . I f l = j + 2 , then d ( v j +1 ) = 2 whenev er v j v l is an chord or not. In this case the co nfiguration G 3 o ccurs since d ( v k ) = 2 , and mov eover, G + v j v l is still pse udo -outerplanar if v j v l 6∈ E ( G ) . So w e assume that l = j + 1 . Now v k , v j , v l form a tria ngle satisfying d ( v k ) = 2 a nd d ( v j ) = 3 . So the c o nfiguration G 2 o ccurs. The case when V [ v j , v l ] is a non-edge can be dealt with simila r ly . Now suppose that V [ v k , v j ] is a non-edg e. B y Cla ims 1–3, b oth V [ v i , v k ] a nd V [ v j , v l ] a r e paths with 1 ≤ k − i ≤ 2 and 1 ≤ l − j ≤ 2 . If k − i = 2 or j − l = 2 , by the s imila r arg umen t as b efore, we either hav e d ( v k − 1 ) = d ( v k ) = 2 or have d ( v j ) = d ( v j +1 ) = 2 , implying that the configuration G 1 o ccurs. So we assume that k − i = l − j = 1 . In this cas e the four vertices v i , v j , v l and v k form a quadrilater al with d ( v i ) = d ( v k ) = 2 , which implies that the configuration G 3 o ccurs in G and furthermore, G + v i v l is still pseudo-o uterplanar if v i v l 6∈ E ( G ) . In the following, we ass ume that V [ v i , v k ] , V [ v k , v j ] and V [ v j , v l ] are all paths, where max { k − i, j − k , l − j } ≤ 2 . Set X = C [ v i , v l ] \{ v i v j , v k v l } a nd x = | X | . It is clea r that x ≤ 3 . Claim 5 . If x = 0 , then G has a su b gr aph isomorphic to one of the c onfigu r ations G 6 – G 11 ; If x = 1 , then G has a sub gr aph isomorphi c one of the c onfigur ations { G 5 , G 12 , G 13 , G 14 } ; If x = 2 , then G has a sub gr aph isomorphic to one of the c onfigur ations { G 5 , G 15 , G 16 } ; If x = 3 , then G has a sub gr aph isomorphi c to the c onfigur ation G 17 . Here, we just show the case when x = 2 a nd v k v j , v j v l ∈ X for example, and leave the discussions ab out other cases to the readers since they are quite similar. In fact, if k − i = 1 (resp. k − i = 2 ), then the configura tion G 15 (resp. G 5 ) would o ccurs in G since d ( v k ) = 4 a nd d ( v i +1 ) = d ( v k +1 ) = d ( v j +1 ) = 2 , and furthermore the drawing of the co nfiguration G 15 (resp. G 5 ) in the figure is just a part of the diag r am of G with its bending edges corresp onding to the chords. Un til no w, Claims 1-5 just complete the pr o of of this theorem for the case when G is 2 -connected. Now we suppo se that G has at leas t t wo blocks. Let B b e an end blo ck and let v 1 , v 2 , · · · , v | B | be the vertices of B that lies in a clo ckwise sequence. Without loss o f g enerality , let v 1 be the unique cut v ertex of B . Claim 6 . B is an out erplanar gr aph. W e prov e that there is no cros sings in B . Supp ose, to the contrary , that there is a chord v i v j that crosses another c hor d v k v l , where 1 ≤ i < k < j < l . No te that the chord v i v j satisfies (1) and (2) now. If it do es not fulfill (3) at this stage. Then there must b e at lea st one pair o f m utually crossed chords contained in either C [ v i , v k ] , or C [ v k , v j ] , or C [ v j , v l ] . W e cho ose one pair v a v b and v c v d among them s uch that a < c < b < d and ther e is no other crossed chords in C [ v a , v d ] b esides v a v b and v c v d . Now s e t i := a , j := b , k := c a nd l := d . Therefor e, in a ny case we can find a pair of mu tually cr ossed chords, v i v j and v k v l , suc h that 1 ≤ i < k < j < l and the three conditions at the be g inning of the pro of are fulfilled. Note that B is an 2-connected pseudo-outerplana r diagram. Thus we can s e t v i , v j , v k , v l as w e did in the 2-co nnected case . Recall the pro ofs of Claims 1-5, ev ery time we find a copy of some config ur ation the vertices v i and v l cannot b e the solid vertices (i.e. the degrees of them in the config uration shall not necessarily to be confirmed). F o r a vertex v ∈ V ( B ) \ { v 1 } , its degree in B is equal to its degree in G , since B is an end blo ck a nd v 1 is the unique cut vertex o f the B . Among the vertices in V [ v i , v l ] , only v i may b e the cut vertex since 1 ≤ i < k < j < l . Therefore, the pro o fs of Cla ims 1-5 a r e also v alid for this claim a nd then the same re s ults would b e obtained. Claim 7 . B has a sub gr aph isomorph ic to one of the c onfigur ations { G 1 , G 2 , G 4 } in such a way that v 1 is not a solid vertex. Since B is a 2 -c o nnected outer pla nar graph, B is hamiltonian. So V [ v 1 , v | B | ] is a path. The pro of o f Claim 2 10 implies that if V [ v i , v k ] is a path with k − i ≥ 3 such that there is no crossed edges in C [ v i , v k ] and no edges b etw een V ( v i , v k ) and V ( v k , v i ) , then G contains one of { G 1 , G 2 , G 4 } in suc h a way that v i and v k are not th e solid v ertices. Thu s in this claim, if | B | ≥ 4 , then w e set i := 1 , k := | B | a nd come bac k to the pro of o f Claim 2 . If | B | ≤ 3 , then it is trivial to s ee that G 1 would appear . This contradiction completes the pro of o f the theorem for the case when G has cut vertices. The following is a straightforw ard corollar y of Theo r em 4.2. Corollary 4. 3. Each pseudo-outerplanar gr aph c ontains a vertex of de gr e e at most 3 . 5 Edge Chromatic Num b er and Linear Arb oricity In this section, we a im to co nsider the problems of cov ering a pseudo-outerpla nar graph G with ∆( G ) ma tc hings or ⌈ ∆( G ) 2 ⌉ linear forests. A gr aph G is χ ′ -critic al if χ ′ ( G ) = ∆( G ) + 1 but χ ′ ( H ) ≤ ∆( G ) for any prop er subgr aph H ⊂ G , is la-critic al if la ( G ) > ⌈ ∆( G ) 2 ⌉ but l a ( H ) ≤ ⌈ ∆( G ) 2 ⌉ for any prop er subgraph H ⊂ G . Lemma 5.1. If G is χ ′ -critic al and uv ∈ E ( G ) , then d ( u ) + d ( v ) ≥ ∆( G ) + 2 . Lemma 5.2. If G is la-critic al and uv ∈ E ( G ) , then d ( u ) + d ( v ) ≥ 2 ⌈ ∆( G ) 2 ⌉ + 2 . The ab ov e tw o lemmas are v ery classic a nd useful; their pro ofs ca n b e found in [3] and [14] res pectively . Given a coloring ϕ of G , c j ( v ) denotes the num b er of edges incident with v co lored by j . Let C i ϕ ( v ) = { j | c j ( v ) = i } , i = 0 , 1 , 2 . Then C 0 ϕ ( v ) ∪ C 1 ϕ ( v ) = { 1 , 2 , · · · , k } if ϕ is a prop er k - e dge-color ing , and C 0 ϕ ( v ) ∪ C 1 ϕ ( v ) ∪ C 2 ϕ ( v ) = { 1 , 2 , · · · , k } if ϕ is a k -tr e e-coloring . F or brevity , in the pro of of Theorem 5.3 w e use the notion k -c oloring to replace the sta temen ts of prop er k - e dge-color ing or k -tr ee-coloring and use the notion PO-gr aph to replace the statement of pseudo-outer planar g r aph. F or a gr aph G a nd t wo distinct vertices u, v ∈ V ( G ) , denote by G + xy the graph obtained from G b y adding an new edge xy if xy 6∈ E ( G ) , or G itself if xy ∈ E ( G ) . Theorem 5.3. L et G b e a pseudo-outerplanar gr aph. If ∆( G ) ≥ 4 , then χ ′ ( G ) = ∆( G ) . Pr o of. Suppose for a co ntradiction that there exists a minimal (in terms of the s ize) pseudo-outerpla na r dia gram G with ∆( G ) ≥ 4 that has no ∆( G ) -coloring. One can easily o bserve that G is 2 -connected and χ ′ -critical. B y Theorem 4.2 and Lemma 5.1, G cont ains at leas t one of the co nfig urations { G 3 , G 4 , G 5 , G 6 , G 12 , G 13 , G 16 , G 17 } . Set S = { 1 , 2 , · · · , ∆( G ) } . If G ⊇ G 3 , then the pseudo-outerplanar graph G ′ = G \{ u, v } admits a ∆( G ) -colo r ing φ by induction hypothesis (when ∆( G ′ ) = ∆( G ) ) or Vizing’s Theorem (when ∆( G ′ ) ≤ ∆( G ) − 1 ). Construct a ∆( G ) -color ing ϕ of G as follows. If C 1 φ ( x ) = C 1 φ ( y ) := L (notice that | L | = ∆( G ) − 2 b y Lemma 5.1), then let ϕ ( ux ) = ϕ ( y v ) ∈ S \ L and ϕ ( uy ) = ϕ ( xv ) ∈ S \ ( L ∪ { ϕ ( ux ) } ) . If C 1 φ ( x ) 6 = C 1 φ ( y ) , then ( S \ C 1 φ ( x )) ∩ C 1 φ ( y ) 6 = ∅ s ince d ( x ) = d ( y ) = ∆( G ) by Lemma 5.1. Let ϕ ( ux ) ∈ ( S \ C 1 φ ( x )) ∩ C 1 φ ( y ) , ϕ ( xv ) ∈ S \ ( C 1 φ ( x ) ∪ { ϕ ( ux ) } ) , ϕ ( v y ) ∈ S \ ( C 1 φ ( y ) ∪ { ϕ ( xv ) } ) and ϕ ( uy ) ∈ S \ ( C 1 φ ( y ) ∪ { ϕ ( y v ) } ) . In each case, w e color the r emain edges of G by the sa me colors used in φ . Th us , we have co nstructed a ∆( G ) -coloring ϕ of G from the ∆( G ) -coloring φ o f G ′ . In the next cases, while constructing a co loring ϕ of G from the c oloring φ of G ′ , w e only give the colorings for the edges in E ( G ) \ E ( G ′ ) , since for every edge e ∈ E ( G ) ∩ E ( G ′ ) w e alwa ys let ϕ ( e ) = φ ( e ) . If G ⊇ G 4 , we shall assume that d ( v ) = d ( w ) = ∆( G ) = 4 beca use of Lemma 5.1. Then the PO-graph G ′ = G \ { x, y , u } admits a 4- c o loring φ . Co ns truct a 4- coloring ϕ of G as follows, where tw o cases are considered without loss of g enerality (wlog . for sho rt). If C 1 φ ( v ) = C 1 φ ( w ) = { 1 , 2 } , then let ϕ ( uy ) = 1 , ϕ ( ux ) = 2 , ϕ ( uw ) = ϕ ( v x ) = 3 and ϕ ( uv ) = ϕ ( wy ) = 4 . If C 1 φ ( v ) = { 1 , 2 } , 1 6∈ C 1 φ ( w ) and 3 ∈ C 1 φ ( w ) , then let ϕ ( uw ) = 1 , ϕ ( ux ) = 2 , ϕ ( xv ) = ϕ ( uy ) = 3 , ϕ ( uv ) = 4 and ϕ ( wy ) ∈ { 2 , 3 , 4 } \ C 1 φ ( w ) . If G ⊇ G 5 , we shall assume that d ( v ) = ∆( G ) = 4 b ecause of Lemma 5.1. Then the PO-graph G ′ = G \ { u } admits a 4-color ing φ . One can ea sily see that ( C 1 φ ( v ) ∩ C 1 φ ( w )) \ { φ ( v w ) } 6 = ∅ , because otherwise v w would be inciden t with four co lors under φ . Assume that C 1 φ ( v ) = { 1 , 2 , 3 } and φ ( v w ) = 3 wlog. If C 1 φ ( w ) 6 = C 1 φ ( v ) , then 11 assume that C 1 φ ( w ) = { 1 , 3 , 4 } wlog. Wherea fter, we can ex tend φ to a 4-co loring of ϕ of G by taking ϕ ( uv ) = 4 and ϕ ( uw ) = 2 . If C 1 φ ( w ) = C 1 φ ( v ) , then we consider t wo s ub cas es. If φ ( xz ) = 4 , then construct a 4-coloring of G b y recolo ring wx and w v with 3 and 4, and coloring uv and uw with 3 and 2, r espe c tiv ely . If φ ( xz ) 6 = 4 , then construct a 4- c o loring of G by recoloring w x with 4 and c oloring uv and uw with 4 and 2, resp ectively . If G ⊇ G 6 , w e shall assume that min { d ( x 0 ) , d ( y 0 ) } ≥ 3 and ∆( G ) = 4 by Lemma 5 .1. Assume first that d ( x 0 ) = d ( y 0 ) = 4 . If x 0 y 0 6∈ E ( G ) , then let N ( x 0 ) = { u, v , x 1 , x 2 } and N ( y 0 ) = { u, v , y 1 , y 2 } . Let G ′ = G \ { u, v } + x 0 y 0 . By Lemma 4.2, the co nfig uration G 6 is a part of the pseudo -outerplanar diagram of G . Thus G ′ can also b e a PO- graph a nd th us G ′ admits a 4-coloring φ b y the minimalit y of G . Set M = { φ ( x 0 x 1 ) , φ ( x 0 x 2 ) , φ ( y 0 y 1 ) , φ ( y 0 y 2 ) } a nd m = | M | . Since the co lors used in φ is at mos t four and x 0 y 0 ∈ E ( G ′ ) , m ≤ 3 (other wise the edge x 0 y 0 cannot be colored under φ because it is already incident with four co lored edges). If m = 3 , assume that φ ( x 0 x 1 ) = φ ( y 0 y 1 ) = 1 , φ ( x 0 x 2 ) = 2 and φ ( y 0 y 2 ) = 3 wlog. Now we can extend φ to a 4 -coloring ϕ of G b y ta k ing ϕ ( uv ) = 1 , ϕ ( vy 0 ) = 2 , ϕ ( ux 0 ) = 3 and ϕ ( v x 0 ) = ϕ ( uy 0 ) = 4 . If m ≤ 2 , as sume that φ ( x 0 x 1 ) = φ ( y 0 y 1 ) = 1 and φ ( x 0 x 2 ) = φ ( y 0 y 2 ) = 2 wlog. No w we can also extend φ to a 4 -coloring ϕ of G by taking ϕ ( uv ) = 1 , ϕ ( v y 0 ) = ϕ ( ux 0 ) = 3 and ϕ ( v x 0 ) = ϕ ( uy 0 ) = 4 . On the other hand, if x 0 y 0 ∈ E ( G ) , let N ( x 0 ) = { u, v , y 0 , x 1 } a nd N ( y ) = { u, v , x 0 , y 1 } . Then x 1 6 = y 1 , otherwise by the 2-connectivity of G we ha ve G ≃ G [ { u , v , x 0 , y 0 , x 1 } ] , which can be 4-color able. Consider the graph G ′ = G \ { u, v } − x 0 y 0 , which admits a 4-co lo ring φ by the minimality of G . If φ ( x 0 x 1 ) = φ ( y 0 y 1 ) = 1 , then let ϕ ( uv ) = 1 , ϕ ( x 0 y 0 ) = 2 , ϕ ( ux 0 ) = ϕ ( vy 0 ) = 3 and ϕ ( vx 0 ) = ϕ ( uy 0 ) = 4 . If φ ( x 0 x 1 ) = 1 and φ ( y 0 y 1 ) = 2 , then le t ϕ ( v y 0 ) = 1 , ϕ ( ux 0 ) = 2 , ϕ ( uv ) = ϕ ( x 0 y 0 ) = 3 and ϕ ( v x 0 ) = ϕ ( uy 0 ) = 4 . Second, as s ume that one of x 0 and y 0 has degree three. Ass ume that d ( x 0 ) = 3 wlog. Let N ( x 0 ) = { u, v , w } . Consider the PO-gra ph G ′ = G − ux 0 . By the minimalit y of G , G ′ has a 4-coloring φ . If A := S \ { φ ( v x 0 ) , φ ( wx 0 ) , φ ( uv ) , φ ( uy 0 ) } 6 = ∅ (recall that S = { 1 , 2 , 3 , 4 } ), then let ϕ ( ux 0 ) ∈ A . Other w is e, as sume that φ ( v x 0 ) = 1 , φ ( w x 0 ) = 2 , φ ( uv ) = 3 and φ ( uy 0 ) = 4 wlog. Since d ( v ) = 3 , φ ( uy 0 ) = 4 and v y 0 ∈ E ( G ′ ) , v is not inciden t with the color 4 under φ . Thus we can e x tend φ to a 4-color ing of G by re c o loring v x 0 with 4 and then coloring ux 0 with 1. If G ⊇ G 12 , we sha ll a ssume that ∆( G ) = 4 b ecause of Lemma 5 .1. Assume firs t that d ( x ) = d ( y ) = 4 . If xy 6∈ E ( G ) , then denote N ( x ) = { v , w , x 1 , x 2 } and N ( y ) = { v , w, y 1 , y 2 } . Co ns ider the graph G ′ = G \ { v , w } + xy + u x + uy . Since the configuration G 12 is a part of the pseudo-outerplanar diagram o f G by Lemma 4.2, we can prop erly add three edg e s xy , ux and uy to G \ { v , w } such that G ′ is still a PO-graph. Th us G ′ admits a 4-color ing φ by the minimality of G . One can see that { φ ( xx 1 ) , φ ( xx 2 ) } 6 = { φ ( y y 1 ) , φ ( y y 2 ) } (otherwise we cannot prop erly co lor the triangle uxy under φ ) and { φ ( xx 1 ) , φ ( xx 2 ) } ∩ { φ ( y y 1 ) , φ ( y y 2 ) } 6 = ∅ (otherwise we cannot co lor the edge xy under φ ). Assume that φ ( xx 1 ) = 1 , φ ( xx 2 ) = φ ( y y 1 ) = 2 a nd φ ( y y 2 ) = 3 wlo g. Then we c a n construct a 4 - coloring ϕ of G by taking ϕ ( uv ) = ϕ ( wy ) = 1 , ϕ ( vw ) = 2 , ϕ ( uw ) = ϕ ( v x ) = 3 and ϕ ( wx ) = ϕ ( v y ) = 4 . If xy ∈ E ( G ) , then deno te N ( x ) = { v , w, y , x 1 } a nd N ( y ) = { v , w , x, y 1 } . W e sha ll also a ssume that x 1 6 = y 1 bec a use otherwise G ≃ G [ { u, v , w, x, y , x 1 } ] b y the 2-connectivity of G , whic h admits a 4-c o loring. Now w e remove u , v and w from the diagram of G . Denote by G ′′ the res ulting diagra m. Then G ′′ is a PO-graph s o that both x and y has degree tw o in G ′′ . Since the diagr am of G minimizes the n umber o f c rossings, xx 1 do es not cross y y 1 in G (and th us in G ′′ ). Denote by G ′ the gr aph obtained from G ′′ b y co n tracting the edge xy . F rom the ab ove arg umen ts, one can s ee that G ′ is still a PO-graph with E ( G ) \ E ( G ′ ) = { uv , u w , v w, v x, w x, v y , w y , xy } . F urther more, by the minimalit y of G , G ′ admits a 4 -coloring φ with φ ( xx 1 ) 6 = φ ( y y 1 ) . Supp ose that φ ( xx 1 ) = 1 and φ ( y y 1 ) = 2 . Then we ca n construct a 4-co lo ring ϕ of G by ta king ϕ ( uw ) = ϕ ( v y ) = 1 , ϕ ( uv ) = ϕ ( wx ) = 2 , ϕ ( v w ) = ϕ ( xy ) = 3 a nd ϕ ( v x ) = ϕ ( w y ) = 4 . Seco nd, as s ume that one of x and y , say x wlog., has degree at most three. If d ( x ) ≤ 2 , then it is ea sy to see that G ≃ G [ { u, v , w, x, y } ] b y the 2-co nnectivit y o f G , which admits a 4-co lo ring. If d ( x ) = 3 , then denote N ( x ) = { v , w, x 1 } . Co ns ider the PO-graph G ′ = G − uv , which admits a 4 -coloring φ by the minimalit y of G . I f A := S \ { φ ( uw ) , φ ( v w ) , φ ( v y ) , φ ( v x ) } 6 = ∅ (recall that S = { 1 , 2 , 3 , 4 } ), then let ϕ ( uv ) ∈ A . Otherwise, assume that φ ( uw ) = 1 , φ ( v w ) = 2 , φ ( v y ) = 3 and φ ( v x ) = 4 wlog . It follows that φ ( wx ) = 3 and φ ( wy ) = 4 . If φ ( xx 1 ) = 1 , then we can co nstruct a 4 -coloring of G by recolor ing v x and uw with 2, recoloring v w with 1 and coloring uv with 4 . If φ ( xx 1 ) = 2 , then we c a n again construct a 4-colo ring o f G by re c o loring v x with 1 and coloring uv with 4. If G ⊇ G 13 , then we shall assume that d ( x ) = ∆( G ) = 4 by Lemma 5.1. Denote the four th neig h bo r o f x by 12 0 x 1 x i x n x 0 y 1 y i y n y w n P u v 2 v 1 w 3 v 2 w 1 v 3 w 2 z 3 z 4 z 5 z 6 z 1 z 7 z 1 u 2 u 3 u … n Q 2 n z … 2 i z 2 i z i u i v i w - 1 Figure 4: Sp ecial pseudo-o uter planar gra phs x 1 and meanwhile ass ume that d ( y ) = 4 and N ( y ) = { v , w , y 1 , y 2 } wlog. Then the PO-graph G ′ = G \ { u, v , w } admits a 4-c o loring φ . Wlog. assume that φ ( xx 1 ) = 1 . Construct a 4-coloring ϕ of G as follo ws. If 1 ∈ C 1 φ ( y ) (suppo s e φ ( yy 1 ) = 1 and φ ( yy 2 ) = 2 wlog.), then let ϕ ( v w ) = 1 , ϕ ( uv ) = ϕ ( wx ) = 2 , ϕ ( vx ) = ϕ ( w y ) = 3 and ϕ ( ux ) = ϕ ( vy ) = 4 . If 1 6∈ C 1 φ ( y ) (suppose φ ( y y 1 ) = 2 and φ ( y y 2 ) = 3 wlo g.), then let ϕ ( vy ) = 1 , ϕ ( ux ) = ϕ ( vw ) = 2 , ϕ ( uv ) = ϕ ( wx ) = 3 and ϕ ( v x ) = ϕ ( w y ) = 4 . If G ⊇ G 16 , then w e s hall assume that d ( x ) = d ( y ) = ∆( G ) = 4 b y Lemma 5.1. Denote the fourth neig hbo r of x and y b y x 1 and y 1 resp ectively . Then the PO-g raph G ′ = G \ { u , v , w, z } a dmits a 4-co loring φ . Construct a 4 - coloring ϕ of G as follows. If φ ( xx 1 ) = φ ( y y 1 ) = 1 , then let ϕ ( v w ) = 1 , ϕ ( ux ) = ϕ ( v z ) = ϕ ( wy ) = 2 , ϕ ( wx ) = ϕ ( v y ) = 3 and ϕ ( uw ) = ϕ ( vx ) = ϕ ( y z ) = 4 . If 1 = φ ( xx 1 ) 6 = φ ( y y 1 ) = 2 , then let ϕ ( v z ) = ϕ ( w y ) = 1 , ϕ ( ux ) = ϕ ( wy ) = ϕ ( v z ) = 2 , ϕ ( wx ) = ϕ ( v y ) = 3 and ϕ ( uw ) = ϕ ( vx ) = 4 . If G ⊇ G 17 , then we shall assume that d ( x ) = d ( y ) = ∆( G ) = 5 b y Lemma 5.1. Then the PO-gra ph G ′ = G \ { u, v , w , z , a } admits a 5- coloring φ . Construct a 5 - coloring ϕ o f G as follows. If C 1 φ ( x ) = C 1 φ ( y ) = { 1 , 2 } , then let ϕ ( uw ) = ϕ ( av ) = 1 , ϕ ( wz ) = ϕ ( uv ) = 2 , ϕ ( xz ) = ϕ ( vw ) = ϕ ( ay ) = 3 , ϕ ( wx ) = ϕ ( v y ) = 4 and ϕ ( v x ) = ϕ ( w y ) = 5 . If | C 1 φ ( x ) ∩ C 1 φ ( y ) | = 1 (suppos e C 1 φ ( x ) = { 1 , 2 } and C 1 φ ( y ) = { 1 , 3 } wlo g.), then let ϕ ( vw ) = 1 , ϕ ( w y ) = ϕ ( av ) = 2 , ϕ ( wz ) = ϕ ( v x ) = 3 , ϕ ( wx ) = ϕ ( uv ) = ϕ ( ay ) = 4 and ϕ ( xz ) = ϕ ( uw ) = ϕ ( v y ) = 5 . If | C 1 φ ( x ) ∩ C 1 φ ( y ) | = 0 (supp ose C 1 φ ( x ) = { 1 , 2 } a nd C 1 φ ( y ) = { 3 , 4 } wlog.), then let ϕ ( v w ) = ϕ ( ay ) = 1 , ϕ ( w z ) = ϕ ( v y ) = 2 , ϕ ( v x ) = ϕ ( uw ) = 3 , ϕ ( wx ) = ϕ ( av ) = 4 and ϕ ( xz ) = ϕ ( uv ) = ϕ ( wy ) = 5 . Theorem 5. 4. F or e ach inte ger n ≥ 1 , ther e exists a 2 -c onn e cte d pseudo-outerplanar G with or der 2 n + 5 and ∆( G ) = 3 so t hat χ ′ ( G ) = ∆( G ) + 1 . Pr o of. Let C = x 0 · · · x n wy n · · · y 0 v ux 0 be a c ycle. W e add edges x i y i for all 1 ≤ i ≤ n and a dd ano ther tw o edges x 0 v and y 0 u to C . Denote the resulting graph by P n (See Figure 4). One can easily chec k that P n is a 2-connected pseudo-outerplanar graph with | P n | = 2 n + 5 and ∆( P n ) = 3 . If P n has a 3 -coloring φ , then we shall hav e φ ( x 0 v ) = φ ( y 0 u ) and φ ( x 0 u ) = φ ( y 0 v ) (otherwise w e canno t color uv prop erly). Ther eby we would deduce that φ ( x i x i +1 ) = φ ( y i y i +1 ) for all 0 ≤ i ≤ n − 1 a nd then φ ( x n w ) = φ ( y n w ) . This final contradiction implies that χ ′ ( P n ) = ∆( P n ) + 1 = 4 . Theorem 5.5. L et G b e a pseudo-outerplanar gr aph. If ∆( G ) = 3 or ∆( G ) ≥ 5 , then l a ( G ) = ⌈ ∆( G ) 2 ⌉ . Pr o of. Since conjecture 1.1 has already b een prov ed for planar g raphs and every PO-graph is plana r (cf. Section 1), this theor em holds trivially when ∆( G ) is o dd. T hus in the following we ass ume that ∆( G ) ≥ 6 and ∆( G ) is even. F or brevit y w e write k = ∆( G ) 2 . Supp ose for a contradiction that there exists a minimal (in terms of the size) 13 pseudo-outerplana r g raph G that has no k -co loring. One ca n ea s ily observe that G is 2 - connected and la-cr itica l. By Theo rem 4 .2 and Lemma 5.2, G co n tains the configur ation G 3 . If xy 6∈ E ( G ) , then by (b) of Lemma 4.2, G ′ = G \ { v } + xy is still a PO-graph. Thus by the minimalit y o f G , G ′ admits a k - coloring φ . Now we can cons truct a k -color ing ϕ of G b y taking ϕ ( v x ) = ϕ ( v y ) = φ ( xy ) and ϕ ( e ) = φ ( e ) for every e ∈ E ( G ) ∩ E ( G ′ ) . If xy ∈ E ( G ) , then consider the PO-graph G ′ = G \ { v } , which has a k - coloring φ by the minimalit y of G . It is easy to s ee that | C 1 φ ( x ) | = | C 1 φ ( y ) | = 1 , since d ( x ) = d ( y ) = ∆( G ) = 2 k b y Lemma 5.2. W e no w construct a coloring ϕ of G by taking ϕ ( v x ) ∈ C 1 φ ( x ) , ϕ ( v y ) ∈ C 1 φ ( y ) and ϕ ( e ) = φ ( e ) for every e ∈ E ( G ) ∩ E ( G ′ ) . If C 1 φ ( x ) 6 = C 1 φ ( y ) , then it is easy to see that ϕ is a k -co lo ring. If C 1 φ ( x ) = C 1 φ ( y ) , then ϕ ( v x ) = ϕ ( vy ) and ϕ is also a k -coloring unless ϕ ( xy ) = ϕ ( v x ) or ϕ ( ux ) = ϕ ( uy ) = ϕ ( v x ) . If ϕ ( x y ) = ϕ ( v x ) , then ϕ ( v x ) 6∈ { ϕ ( ux ) , ϕ ( uy ) } and thus we can e xchange the color s on ux and v x . One can easy to chec k that the resulting color ing of G is a k -colo ring. If ϕ ( ux ) = ϕ ( uy ) = ϕ ( v x ) , then w e r ecolor xy with ϕ ( v x ) and recolor b oth v x and uy with ϕ ( xy ) . The r esulting coloring of G is also a k -coloring. Theorem 5.6. F or e ach inte ger m ≥ 1 , ther e exists a 2 -c onne cte d pseudo-outerplanar G with or der 10 m + 5 and ∆( G ) = 4 so t hat l a ( G ) = ⌈ ∆( G ) 2 ⌉ + 1 . Pr o of. Let C = z 1 · · · z 2 n z 1 be a cycle and T i = u i v i w i u i (1 ≤ i ≤ n ) b e triangles. Supp ose that they a re pairwise disjoint . No w for each 1 ≤ i ≤ n , add fours edges v i z 2 i − 1 , v i z 2 i , w i z 2 i − 1 and w i z 2 i . Denote the res ulting gr aphs b y Q n (See Figure 4 ). One can easily chec k that Q n is a 2-co nnected pseudo-outerplanar graph with ∆( Q n ) = 4 . Consider the g raph Q 2 m +1 ( m ≥ 1 ). It is trivial that | Q 2 m +1 | = 10 m + 5 and la ( Q 2 m +1 ) ≤ 3 b y Lemma 5 .2. If Q 2 m +1 has a 2-colo r ing φ , then we shall ha ve φ ( z 2 i − 2 z 2 i − 1 ) 6 = φ ( z 2 i z 2 i +1 ) for all 1 ≤ i ≤ 2 m + 1 , where z 0 = z 4 m +2 and z 4 m +3 = z 1 (otherwise we ca nnot prop erly color the set of edges { u i v i , v i w i , w i u i , v i z 2 i − 1 , v i z 2 i , w i z 2 i − 1 , w i z 2 i } for some i ). Howev er, the size of the set { z 2 z 3 , z 4 z 5 , · · · , z 4 m +2 z 1 } is 2 m + 1 , which is o dd, but there are only tw o colors that ca n b e used in φ . This final contradiction implies that l a ( Q 2 m +1 ) = ⌈ ∆( Q 2 m +1 ) 2 ⌉ + 1 = 3 . 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