Group edge choosability of planar graphs without adjacent short cycles

Group edge choosability of planar graphs without adjacen t short c yc les ∗ Xin Zhang and Guizhen Liu † School of Mathematics , Shandon g U ni ve rsity , Jinan 2501 00, P . R. China Abstract In this pape r , we aim to introd uce the group ve rsion of edge colori ng and list edge coloring, and prov e that all 2-degen erate graphs along with some planar graphs without adjacent short cycles is group ( ∆ ( G ) + 1 )-edge - choos able while some pl anar graphs with large girth and maximum deg ree is group ∆ ( G )-edge-choo sable. Ke ywords : group edge colorin g, list colori ng, planar graphs, short cyc les. MSC : 05C15, 05C20. 1 Introduction All graphs considered in this paper are finite, simp le and undirected. W e use V ( G ), E ( G ), δ ( G ) and ∆ ( G ) to denote the vertex set, the edge set, the m inimum degree and the maxim um degree of a graph G . By d G ( v ), w e denotes the degree o f v in G . For a plane graph G , F ( G ) d enotes its fac e set and d G ( f ) denot es the degree of a face f in G . The girth g ( G ) of a graph G is the length of its sm allest cycle or + ∞ if G is a forest. Throughout this p aper , a k -, k + - and k − -verte x ( resp. face) is a verte x (resp. face) of de gree k , at least k and at mos t k . An i -alternating cycles in a graph G is a cycle of e ven length i n which alternate v ertices have degree i . W e say Email addresses: sdu.zh ang@yaho o.com. cn (X. Zhang), gzliu@sdu.e du.cn (G. Liu) ∗ This research is suppo rted by NSFC (1 09711 21, 610 7023 0), RFDP(20100 13112 0017 ) an d GIIFSDU (yzc1 0040 ). † Correspon ding author . 1 a graph G is k -degenerate if δ ( H ) ≤ k for every subgraph H ⊆ G . Any u ndefined notation follows that of Bondy and Murty [ 1 ]. In 19 92, Jaeger et al. [ 5 ] introduced a concept of group connectivity as an generalization of nowhere zero flows and its dual con cept group colorin g. They proposed t he definition of group colorability of graphs as the equiv alence of group connectivity of M , where M is a cographic matroid. Let G be a graph and A b e an Abelian group . Denote F ( G , A ) to be the set of all functi ons f : E ( G ) 7→ A and D to be an arbit rary orientati on of E ( G ). W e say G is A -colorable under the orientation D if for any fu nction f ∈ F ( G , A ), G has an ( A , f )-colorin g, n amely , a vertex coloring c : V ( G ) 7→ A such that c ( u ) − c ( v ) , f ( u v ) for every directed edge u v from u to v . In [ 8 ], Lai and Zhang presented that for any Abelian group A , a graph G is A -colorable under the orientation D if and only if G i s A -colorable under e very orientation of E ( G ). That is to say , the group colorability of a graph is independent of the ori entation of E ( G ). The group chromatic num ber of a graph G , denot ed by χ g ( G ), is d efined to be the m inimum m for which G is A -colorable for any Abelian group A of order at least m . Clearly , χ ( G ) ≤ χ g ( G ), where χ ( G ) is the chromatic num ber of G . Lai and Zh ang [ 9 ] proved that χ g ( G ) ≤ 5 for e very planar graph G and Kr ´ al’ et al. [ 6 ] constructed a planar graph with the group chromatic numb er five. This implies the well-known Four -Colors Theorem for ordinary coloring s can not be extended t o group colorings. Nevertheless, some theorems for o rdinary vertex colorings, such as Brooks’ Theorem, stil l can be extended. T he following theorem is due to Lai et al. [ 10 ]. Theor em 1.1. For any connected simple graph G , χ g ( G ) ≤ ∆ ( G ) + 1, where equality holds if and only if G is either a cycle or a complete graph. Here notice t hat for an ev en cycle C 2 n , we have χ g ( C 2 n ) = 3 by Theorem 1.1 but χ ( C 2 n ) = 2. In 2 004, Kr ´ al’ and Nejedl ´ y [ 7 ] con sidered list g roup colori ng as an extension of list coloring and group coloring. Let G be a graph, A be an Abelian group of order at l east k and L : V ( G ) 7→ 2 A be a k -uniform lis t assignment of V ( G ). Denote F ( G , A ) to be t he set of all functions f : E ( G ) 7→ A and D to be an arbitrary orientation of E ( G ). W e say G is group k -choosable under the orientation D if for any function f ∈ F ( G , A ), G has an ( A , L , f )-coloring, t hat is an ( A , f )-colori ng c such that c ( v ) ∈ L ( v ) for e very v ∈ V ( G ). Note that the choice of an o rientation of edges of G is ei ther not essential in this definition. The group choice num ber of a graph G , denot ed by χ g l ( G ), is defined t o be the minim um k fo r which G is group k -choos able. In [ 7 ], th e a uthors showed that χ g l ( G ) = 2 if and o nly if G is a forest. Omidi [ 11 ] prove d the group choice number of a graph without K 5 -minor or K 3 , 3 - minor and wit h girth at least 4 (resp. 6) i s at m ost 4 (resp. 3). In [ 2 ], Chuang 2 et al. also established the g roup choos ability version of Brooks’ Theorem, which extends Theorem 1.1 . Theor em 1.2. For any connected simp le graph G , χ g l ( G ) ≤ ∆ ( G ) + 1, where equality holds if and only if G is either a cycle or a complete graph. In this paper , we aim to introduce a group version of edge coloring and list edge coloring. Reca ll that the line graph of a graph G , denoted by L ( G ), is a graph such t hat each vertex of L ( G ) represents an edge of G and two vertices of L ( G ) are adjacent i f and o nly if their corresponding edges share a common endpoint in G . For an edge u v ∈ E ( G ), we use e u v to denot e t he verte x in L ( G ) that represents u v in G . Clearly , th e edge chromatic number χ ′ ( G ) o f a graph G i s equal to the vertex chromatic num ber χ ( L ( G )) of it s l ine graph L ( G ). In view of this, the group version of edge coloring and list e dge c oloring can b e defined naturally . For an Abelian group A of ord er at l east k , w e say G is g roup A -edge-colorable if L ( G ) is group A -colorable and say G is group k -edge-choosable i f L ( G ) is group k -choos able. By χ ′ g ( G ) = χ g ( L ( G )) and χ ′ g l ( G ) = χ g l ( L ( G )), we denotes the group edge chrom atic num ber and the group edge choice number of a graph G . First of all, we hav e the following basic theorem. Theor em 1.3. For an y connected simple graph G , ∆ ( G ) ≤ χ ′ g ( G )            = χ ′ g l ( G ) = 2 , if G i s a path; = χ ′ g l ( G ) = 3 , if G i s a cycle; ≤ χ ′ g l ( G ) ≤ 2 ∆ ( G ) − 2 , if ∆ ( G ) ≥ 3. Pr oof. Since χ ′ g l ( G ) ≥ χ ′ g ( G ) ≥ χ ′ ( G ) ≥ ∆ ( G ), t he left inequali ty in above theo- rem holds. If G is a path (resp. cycle), then L ( G ) is also a p ath (resp. cycle). So by Theorems 1.1 and 1.2 , we h a ve χ ′ g ( G ) = χ ′ g l ( G ) = 2 (resp . 3). If ∆ ( G ) ≥ 3 , then G is neither a cycle nor a star , which implies L ( G ) is neit her a cycle nor a complete graph. So χ ′ g ( G ) ≤ χ ′ g l ( G ) = χ g l ( L ( G )) ≤ ∆ ( L ( G )) ≤ 2 ∆ ( G ) − 2 by Theorems 1.1 and 1.2 .  From Theorem 1.3 , we can find that χ ′ g ( G ) ≤ χ ′ g l ( G ) ≤ ∆ ( G ) + 1 for ev ery graph with maxi mum degree 3 and χ ′ g ( G ) = χ ′ g l ( G ) for e very graph with maximum de- gree 2. These evidences motiv ate us to conjecture the analogue of V izing’ s Theo- rem o n edge chromatic numb er and l ist edg e col oring Conjecture on edge cho ice number . Conjectur e 1.4. For an y simp le graph G , ∆ ( G ) ≤ χ ′ g ( G ) ≤ ∆ ( G ) + 1. Conjectur e 1.5. For an y simp le graph G , χ ′ g ( G ) = χ ′ g l ( G ). 3 In t he ne xt section, we will confirm C onjecture 1.4 for all 2-degenerate graphs and some planar graphs without adjacent short c ycles and confirm Conjecture 1.5 for some planar graphs with large girth a nd maximum degree. For a nonnegati ve integer i , we call a graph G is group ( ∆ ( G ) + i )-edge-critical if χ ′ g l ( G ) > ∆ ( G ) + i but χ ′ g l ( H ) ≤ ∆ ( H ) + i for e very proper subgraph H ⊂ G . The ( ∆ ( G ) + i )-edge-critical graph in terms of l ist edge coloring can be d efined similarly . In most of the articles concerning list ( ∆ + 1)-edge coloring of p lanar graphs i n th e lit erature s uch as [ 3 ] and [ 4 ], it was proved and ess ential that a 3- alternating cycle C can not appear i n a ( ∆ + 1 )-edge-critical graph G because i f such a cycle C do exist, then G − E ( C ) i s ( ∆ + 1 )-edge choos able and e very edge of C has at least two av ailable colo rs since it is in cident with ∆ ( G ) + 1 edges, of which ∆ ( G ) − 1 are colored, which i mplies t hat one can extend the list ( ∆ + 1)- edge coloring of G − E ( C ) to G by the fa ct that e ven c ycles are 2-edge-choosable. Howe ver , this t echnique is in valid for group edge choosabilit y since any cycle is not group 2-edge-choosable by Theorem 1.3 . 2 Main r esult s and their pr oof s W e be gin with this section by pro ving an useful Lemma, w hich will be frequently used i n the next proofs and im plies Conjecture 1.4 holds for all 2-degenerate graphs. Lemma 2.1. Let i be a nonnegative in teger and G be a group ( ∆ ( G ) + i )-edge- critical graph. Then G is connected and d G ( u ) + d G ( v ) ≥ ∆ ( G ) + i + 2 for an y edge u v ∈ E ( G ). Pr oof. Th e connectivity of G direc tly follows from its definition. Suppose there is an edge u v ∈ E ( G ) such that d G ( u ) + d G ( v ) ≤ ∆ ( G ) + i + 1. Th en for an Abeli an grou p A of order at least ∆ ( G ) + i , a ( ∆ ( G ) + i )-uniform list ass ignment L : V ( L ( G )) 7→ 2 A and a function f ∈ F ( L ( G ) , A ), L ( G ) is not ( A , L , f )-colorable but L ( G − u v ) is . Let c be an ( A , L , f )-coloring of L ( G − u v ). Noti ce that now in L ( G ) the on ly uncolored verte x under c is e u v , whi ch is adjacent to m = d G ( u ) + d G ( v ) − 2 ≤ ∆ ( G ) + i − 1 colored vertices, say e 1 , e 2 , · · · , e m . W i thout any loss of generality we ass ume e u v is the head of each edge e i e u v in L ( G ) under a giv en orientati on D of E ( L ( G )), where 1 ≤ i ≤ m . Now assi gn e u v a color in S = L ( e u v ) − S m i = 1 { c ( e i ) − f ( e i e u v ) } . Notice that | S | ≥ ∆ ( G ) + i − m ≥ 1. So we ha ve e xtended c to an ( A , L , f )-coloring of L ( G ). Thi s implies G i s group ( ∆ ( G ) + i )-edge-choosabl e, a contradiction.  Corollary 2.2. Let i be a nonnegativ e integer and G be a grou p ( ∆ ( G ) + i )-edge- critical graph. Then δ ( G ) ≥ i + 2 . 4 Corollary 2.3. Every 2-degenera te graph is group ( ∆ ( G ) + 1)-edge-choosable. Theor em 2.4. Let G be a planar g raph such t hat G does not con tain an i -cycle adjacent to a j -cycle where 3 ≤ i ≤ s and 3 ≤ j ≤ t . If (1) s = 3, t = 3 and ∆ ( G ) ≥ 8, or (2) s = 3, t = 4 and ∆ ( G ) ≥ 6, or (3) s = 4, t = 5 and ∆ ( G ) ≥ 5, or (4) s = 4, t = 7, then G is group ( ∆ ( G ) + 1)-edge-choosable. Pr oof. Th e proof i s carried out by contradiction and discharging. Suppos e G is a mi nimum coun terexample to the theorem. Then by Lemma 2.1 , one can easily find that G is a connected and group ( ∆ ( G ) + 1)-edge-critical planar graph with δ ( G ) ≥ 3. By Euler’ s F orm ula, for any n > 2 m > 0, we have X v ∈ V ( G ) [( n 2 − m ) d G ( v ) − n ] + X f ∈ F ( G ) ( md G ( f ) − n ) = − 2 n < 0 . (2.1) Assign each vertex v ∈ V ( G ) an initial charge c ( v ) = ( n 2 − m ) d G ( v ) − n and each face f ∈ F ( G ) an initial char ge c ( f ) = md G ( f ) − n . Then by ( 2.1 ), we hav e P x ∈ V ( G ) ∪ F ( G ) c ( x ) < 0. T o prove t he theorem, we are ready to construct a ne w char g e function c ′ on V ( G ) ∪ F ( G ) according some defined dis char ging rules, which o nly move charge aroun d but do not a ff ect the total charges, so that after discharging the final char ge c ′ ( x ) of each element x ∈ V ( G ) ∪ F ( G ) is nonnegative. This contradiction completes the proof of the theorem in final. In th e following, we call a f ace f ∈ F ( G ) is simple if the bound ary of f is a cycle and denote m v ( f ) to be the num ber of t imes through v by a face f in clockwi se order . Obviously , if v is a non-cut verte x or f i s a simple face , then m v ( f ) = 1. (1) Let S be the s et of 3-vertices, 4-vertices and 5-vertices i n G . By Lem ma 2.1 , we can claim that S forms an independent set in G sin ce ∆ ( G ) ≥ 8. Now we choose m = 2 and n = 6 in ( 2.1 ) and define the dischar ging rules as follows: R1.1 . From each 4 + -face f to i ts incident verte x v ∈ S , transfer m v ( f ). R1.2 . From each 8 + -verte x u to its adjacent 3-vertex v , transfer 1 2 if u v is incident with a 3-cycle. W ithout any loss of g enerality , we always assume v is a n on-cut vertex and f is simple i n the following arguments (because during the calculational part of discharging, the case when v is a cut vertex th at is incident with a non-simple face f is equivalent to the case when v i s i ncident with m v ( f ) si mple face s with the same degree o f f , and the case when f i s a non-simple face that is i ncident with a cut verte x v is equiv alent to the case when f is incident with m v ( f ) non-cut 5 vertices with the same d egree of v ). Supp ose d G ( v ) = 3. Then by Lem ma 2.1 , v is adjacent to t hree 8 + -vertices. If v i s incid ent with a 3-face, then v i s also incid ent with two 4 + -faces since there are no t wo adjacent 3-cycles in G . Th is imp lies c ′ ( v ) ≥ c ( v ) + 2 × 1 2 + 2 × 1 = 0 b y R1.1 and R1.2. If v is not incident with any 3-faces, then c ′ ( v ) ≥ c ( v ) + 3 × 1 = 0 by R1.1. Suppose 4 ≤ d G ( v ) ≤ 5. On e can easy show that v is incident w ith a t least two 4 + -faces, which impl ies c ′ ( v ) ≥ c ( v ) + 2 × 1 = 0 . Suppose 6 ≤ d G ( v ) ≤ 7. Then it is easy t o see w ′ ( v ) = w ( v ) ≥ 0. Suppo se d G ( v ) ≥ 8. Notice t hat any two 3-cycles are not adjacent in G , so G is incident with a t most ⌊ d G ( v ) 2 ⌋ 3-faces, which i mplies v m ay transfer charges to at most ⌊ d G ( v ) 2 ⌋ 3-vertices by R1.2 s ince any 3-vertices are not adjacent in G either . So we have c ′ ( v ) ≥ d G ( v ) − 6 − 1 2 ⌊ d G ( v ) 2 ⌋ ≥ 0 for d G ( v ) ≥ 8. Suppose d G ( f ) = 3. Then it i s trivial that c ′ ( f ) = c ( f ) = 0. Suppose d G ( f ) ≥ 4. Then f may t ransfer charges to at m ost ⌊ d G ( f ) 2 ⌋ vertices by R1.1 since S is an independent set in G . This i mplies c ′ ( f ) ≥ 2 d G ( f ) − 6 − ⌊ d G ( f ) 2 ⌋ ≥ 0 for d G ( f ) ≥ 4 in final. (2) W e choos e m = 3 and n = 10 in ( 2.1 ) and define the discharging rules as follows: R2.1 . From each 6 + -verte x to its adjacent 3-verte x , transfer 1 3 . R2.2 . From each 4-face f to its incident verte x v , transfer m v ( f ) if d G ( v ) = 3, 1 2 m v ( f ) if d G ( v ) = 4. R2.3 . From each 5 + -face f to its incident vertex v , transfer 3 2 m v ( f ) i f d G ( v ) = 3, m v ( f ) if d G ( v ) = 4. R2.4 . From each 5 + -face to its adjacent 3-face , transfer 1 3 . Suppose d G ( v ) = 3. Then by Lemma 2.1 , v is adjacent t o t hree 6 + -vertices s ince ∆ ( G ) ≥ 6. If v is incident wi th a 3-face, then v is also inci dent with two 5 + -face by the condition in the theorem. This im plies c ′ ( v ) ≥ c ( v ) + 3 × 1 3 + 2 × 3 2 = 0 by R2.1 and R2.3. If v is i ncident with no 3-faces, then v is incident with three 4 + -face, w hich impl ies c ′ ( v ) ≥ c ( v ) + 3 × 1 3 + 3 × 1 = 0 by R2.1, R2.2 and R2.3. Suppose d G ( v ) = 4. If v is incident with a 3-face, then v is incident with at least two 5 + -faces, which implies c ′ ( v ) ≥ c ( v ) + 2 × 1 = 0 by R2.3. If v is incident wi th no 3-faces, then v is incident wi th four 4 + -faces, which implies c ′ ( v ) ≥ c ( v ) + 4 × 1 2 = 0 by R2.2 and R2.3. Suppos e d G ( v ) = 5. Then it is easy to see c ′ ( v ) = c ( v ) = 0. Suppose d G ( v ) ≥ 6. Then by R2.1, we ha ve c ′ ( v ) ≥ 2 d G ( v ) − 10 − 1 3 d G ( v ) ≥ 0. Suppose d G ( f ) = 3. Then by t he condi tion of the theorem f is adj acent to three 5 + -faces, impl ying c ′ ( f ) ≥ c ( f ) + 3 × 1 3 = 0 by R2.4. Suppose d G ( f ) ≥ 4. Then f i s incident with at most ⌊ d G ( f ) 2 ⌋ 4 − -vertices since there is no adjacent 4 − -vertices in G by Lemma 2.1 . T his implies c ′ ( f ) ≥ c ( f ) − 2 × 1 = 0 for d G ( f ) = 4 by R2.2, and c ′ ( v ) ≥ 3 d G ( f ) − 10 − 1 3 d G ( f ) − 3 2 ⌊ d G ( f ) 2 ⌋ > 0 for d G ( f ) ≥ 5 by R2.3 and R2.4. (3) W e choose m = 2 and n = 6 in ( 2.1 ) and define the discharging rules as follows: 6 R3.1 . From each 5-face f to its incident verte x v , transfer m v ( f ) if d G ( v ) = 3, 1 2 m v ( f ) if d G ( v ) = 4, 1 5 m v ( f ) if d G ( v ) = 5. R3.2 . From each 6 + -face f to its incident vertex v , transfer 3 2 m v ( f ) i f d G ( v ) = 3, m v ( f ) if d G ( v ) = 4, 1 3 m v ( f ) if d G ( v ) = 5. Suppose d G ( v ) = 3. If v is incident with a 4 − -face, then v is also incid ent with two 6 + -faces by the conditi on of the theorem, which implies by R3.2 that c ′ ( v ) ≥ c ( v ) + 2 × 3 2 = 0. If v is incident with no 4 − -faces, then by R3.1 and R3.2 we ha ve c ′ ( v ) ≥ c ( v ) + 3 × 1 = 0. Suppose d G ( v ) = 4. If v is incident w ith a 4 − -face, then v is incident wit h at least two 6 + -faces, wh ich implies c ′ ( v ) ≥ c ( v ) + 2 × 1 = 0 by R3.2. If v is inci dent wi th n o 4 − -faces, then by R3.1 and R3.2 we also hav e c ′ ( v ) ≥ c ( v ) + 4 × 1 2 = 0 . Suppo se d G ( v ) = 5. If v is inci dent wit h at least one 4 − - face, then v is incident with e ither three 6 + -faces implying c ′ ( v ) ≥ c ( v ) + 3 × 1 3 = 0 by R3.2, or two 5 + -faces and tw o 6 + -faces imp lying c ′ ( v ) ≥ c ( v ) + 2 × 1 5 + 2 × 1 3 > 0 by R3.1 and R3.2. If v is incident wit h no 4 − -faces, then by R3.1 and R3.2 we sti ll hav e c ′ ( v ) ≥ c ( v ) + 5 × 1 5 = 0. Suppose d G ( v ) ≥ 6 or 3 ≤ d G ( f ) ≤ 4. Then it i s clear that c ′ ( v ) = c ( v ) ≥ 0 and c ′ ( f ) = c ( f ) ≥ 0. Suppose d G ( f ) = 5. If f is incident with no 3-v erti ces, then by R3.1 we have c ′ ( f ) ≥ c ( f ) − 5 × 1 2 > 0. If f i s incident with at leat one 3-v ertex, n ote that any 3-verte x ca n not be adjacent to a 4 − -verte x in G by Lemma 2.1 , so f is also in cident with at least two 5 + -vertices. This impli es c ′ ( f ) ≥ c ( f ) − 2 × 1 5 − 3 × 1 > 0 by R3.1. Supp ose d G ( f ) ≥ 6. Th en we shall hav e d G ( f ) − n 3 − n 4 ≥ n 3 by Lemma 2.1 since ∆ ( G ) ≥ 5, w here n i denotes the number of i -vertices that are incident with f i n G . Thi s impl ies b y R3.2 t hat c ′ ( f ) ≥ 2 d G ( f ) − 6 − 3 2 n 3 − n 4 − 1 3 ( d G ( f ) − n 3 − n 4 ) = d G ( f ) − 6 − 2 3 (2 n 3 + n 4 − d G ( f )) + 1 6 n 3 ≥ 0 in final. (4) W e shall assume ∆ ( G ) ≥ 4 in this part because the cases when ∆ ( G ) ≤ 3 hav e been p roved in Theorem 1.3 . No w we als o choose m = 2 and n = 6 i n ( 2.1 ) and define the discharging rules as follo ws: R4.1 . From each face f of d egree between 5 and 7 to its i ncident vertex v , transfer m v ( f ) if d G ( v ) = 3, 1 2 m v ( f ) if d G ( v ) ≥ 4. R4.2 . From each 8 + -face f to its incident vertex v , transfer 3 2 m v ( f ) i f d G ( v ) = 3, m v ( f ) if d G ( v ) ≥ 4. Note t hat the above dis char ging rules are highly sim ilar to the o nes in part (3). So by a same analysis as in the previous part, one can also check that c ′ ( v ) ≥ 0 for all v ∈ V ( G ) and c ′ ( f ) ≥ 0 for 3 ≤ d G ( f ) ≤ 4. Now we s hall only consider 5 + -faces. Note that any 3-vertices can n ot be adjacent in G by Lemma 2.1 because we ha ve a lready assumes ∆ ( G ) ≥ 4. Thus n 3 ≤ ⌊ d G ( f ) 2 ⌋ for any f ∈ F ( G ), where n 3 is defined simi larly as in part (3). Suppose 5 ≤ d G ( f ) ≤ 7. Then b y R4.1, we can deduce that c ′ ( f ) ≥ 2 d G ( f ) − 6 − n 3 − 1 2 ( d G ( f ) − n 3 ) ≥ 3 2 d G ( f ) − 6 − 1 2 ⌊ d G ( f ) 2 ⌋ ≥ 0. Suppose d G ( f ) ≥ 8. W e still have c ′ ( f ) ≥ 2 d G ( f ) − 6 − 3 2 n 3 − 1 × ( d G ( f ) − n 3 ) ≥ d G ( f ) − 6 − 1 2 ⌊ d G ( f ) 2 ⌋ ≥ 0 by R4.2 in final. Thi s completes the proof of the 7 theorem.  As an immediately corollary of Theorem 2.4 , we have the following two re- sults. Corollary 2.5. Every planar graph with girth g ( G ) ≥ 5 is group ( ∆ ( G ) + 1)-edge- choosable. Corollary 2.6. Every planar graph with girth g ( G ) ≥ 4 and maxim um degree ∆ ( G ) ≥ 6 is group ( ∆ ( G ) + 1)-edge-choosable. Another interesting topic concerting group edge colorings and list group edge colorings is to determine which class of graphs sati sfies χ ′ g ( G ) = χ ′ g l ( G ). In v iew of this, we end this paper by proving the following theorem, which confirms Con- jecture 1.5 for some planar graphs with large girth and maximum de gree. Theor em 2.7. Let G be a pl anar graph with maximum degree ∆ ( G ) ≥ ∆ ≥ 3. If g ( G ) ≥ 4 + ⌈ 8 ∆ − 2 ⌉ , then χ ′ g ( G ) = χ ′ g l ( G ) = ∆ ( G ). Pr oof. W e just need t o prove χ ′ g l ( G ) = ∆ ( G ) here. Suppo se, to the contrary , that G is a group ∆ ( G )-edge-critical graph. Let c ( v ) = 2 d G ( v ) − 6 if v ∈ V ( G ) and c ( f ) = d G ( f ) − 6 if f ∈ V ( G ). Then by ( 2.1 ), we have P x ∈ V ( G ) ∪ F ( G ) c ( x ) < 0. Now we redistri bute the charge of t he vertices and faces of G according t he following discharging rules: R1 . From each vertex of m aximum degree to its adjacent 2-verte x, t ransfer 2 − 6 ∆ . R2 . From each face f to its incident 2-vertex v , transfer ( 6 ∆ − 1) m v ( f ). W e shall get a contradiction by proving c ′ ( x ) ≥ 0 for eve ry x ∈ V ( G ) ∪ F ( G ), where c ′ ( x ) is t he final charge of the element x after d ischarging. Supp ose d G ( v ) = 2. Then by Lemma 2.1 , the two neighbors of v shall be both ∆ ( G )-vertices, which implies c ′ ( v ) ≥ c ( v ) + 2 × (2 − 6 ∆ ) + 2 × ( 6 ∆ − 1) = 0 by R1 and R2. Suppose 3 ≤ d G ( v ) ≤ ∆ ( G ) − 1 (if exists). Then it is clear that c ′ ( v ) = c ( v ) ≥ 0. Suppose d G ( v ) = ∆ ( G ). Then by R1, one can easil y d educe that c ′ ( v ) ≥ 2 ∆ ( G ) − 6 − ∆ ( G )(2 − 6 ∆ ) ≥ 0 since ∆ ( G ) ≥ ∆ . Suppos e f i s a face i n G . Similarly as in t he proof of Theorem 2.4 , with out loss of generality , we can assume f is s imple. Then by Lemma 2.1 , f is incident with at most ⌊ d G ( f ) 2 ⌋ 2 -vertices. This i mplies by R2 that c ′ ( f ) ≥ d G ( f ) − 6 − ( 6 ∆ − 1) ⌊ d G ( f ) 2 ⌋ ≥ 3 ∆ − 6 2 ∆ g ( G ) − 6 ≥ 3 ∆ − 6 2 ∆ · 4 ∆ ∆ − 2 − 6 = 0 in fi nal.  References [1] J. A. Bondy and U. S. R. Murty , Graph Theory with Appl ications, North- Holland, New Y ork, 1976. 8 [2] H. Chuang, H.-J. Lai, G . R. Omidi and N. Zakeri, On group choosabilit y of graphs I, manuscript submitted. [3] N. Cohen and F . Hav et, Planar graphs with maxim um degree ∆ ≥ 9 are ( ∆ + 1)-edge-choosable: A short proof, INRIA-00432389 (2009), a vailable on line at http: // hal.inri a.fr / inria-00432389 / PDF / RR-7098.pdf. [4] J. Hou, G. Liu and J. Cai, Edge-choosabil ity of pl anar graphs wit hout adj a- cent triangles or without 7-cycles, Discrete Mathematics 309 (2009) 77–84. [5] F . J aeger , N. Lini al, C. Payan and M . T arsi, Group connectivity of graphs— A nonh omogeneous analo gue of nowhere-ze ro flo w properties, J. Combin. Theory Ser . B 56 (1992) 165–182. [6] D. Kr ´ al’, O. P angr ´ ac and H.-J. V oss, A Note on Group Colorings, Journal of Graph Theory 50 (2005) 123–129. [7] D. Kr ´ al’ and P . Nejedl ´ y, Grou p coloring and list g roup coloring are Π P 2 - complete, Lecture Notes in Computer Science 3153 (2004) 274–287. [8] H.-J. Lai and X.-K. Zhang, Group Colorability of Graphs, Ars Combinatoria 62 (2002) 299–317. [9] H.-J. Lai and X .-K. Zhang, Group Chromat ic Number of Graphs without K 5 -minors, Graphs and Combinatori cs 18 (2002) 147–154. [10] H.-J. Lai, X. Li and G. Y u, An inequali ty for the group chromatic number of a graph, Discrete Mathematics 307 (2007) 3076–3080. [11] G. R. Omidi , A note on group cho osability of g raphs with girth at least 4, Graphs and Combinatorics (2010) Doi: 10 .1007 / s00373-010 -0971-4. 9