Total coloring of pseudo-outerplanar graphs

T otal coloring of pseudo-outerplanar graphs ∗ Xin Zhang, Guizhen Liu † School of Mathematics, Shandong Univ ersity , Jinan 250100, China Abstract A graph is pseudo-outerplanar if each of its blocks has an embedding in the plane so that the vertices lie on a fix ed circle and the edges lie inside the disk of this circle with each of them crossing at most one another . In this paper, the total coloring conjecture is completely confirmed for pseudo- outerplanar graphs. In particular , it is pro ved that the total chromatic number of ev ery pseudo-outerplanar graph with maximum degree ∆ ≥ 5 is ∆ + 1. Ke ywords: pseudo-outerplanar graph, total coloring, maximum degree. 1 Intr oduction A total coloring of a graph G is an assignment of colors to the vertices and edges of G such that every pair of adjacent / incident elements recei ve di ff erent colors. A k -total coloring of a graph G is a total coloring of G from a set of k colors. The minimum positi ve inte ger k for which G has a k -total coloring, denoted by χ 00 ( G ), is called the total chromatic number of G . It is easy to see that χ 00 ( G ) ≥ ∆ ( G ) + 1 for any graph G by looking at the color of a v ertex with maximum degree and its incident edges. The next step is to look for a Brooks-typed or V izing-typed upper bound on the total chromatic number in terms of maximum de gree. It turns out that the total coloring version of maximum degree upper bound is a di ffi cult problem and has eluded mathematicians for nearly 50 years. The most well-kno wn speculation is the total coloring conjecture, independently raised by Behzad [1] and V izing [3], which asserts that e very graph of maximum degree ∆ admits a ( ∆ + 2)-total coloring. This conjecture remains open, howe ver , man y beautiful results concerning it have been obtained (cf. [5]). In particular , the total chromatic number of all outerplanar graphs has been determined completely by Zhang et al. [7] and that of all series-parallel graphs [4] has been determined completely by W u and Hu [7]. Email addresses: sdu.zhang@yahoo.com.cn, gzliu@sdu.edu.cn. ∗ Supported in part by NSFC(10971121, 11101243, 61070230), RFDP(20100131120017) and GI- IFSDU(yzc10040). † Corresponding author . 1 A graph is pseudo-outerplanar if each of its blocks has an embedding in the plane so that the vertices lie on a fixed circle and the edges lie inside the disk of this circle with each of them crossing at most one another . For example, K 2 , 3 and K 4 are both pseudo-outerplanar graphs. This notion was introduced by Zhang, Liu and W u in [6], where the edge-decomposition of pseudo-outerplanar graphs into forests with a specified property was studied. In this paper , we prov e that the total chromatic number of e v ery pseudo-outerplanar graph with maximum de gree ∆ ≥ 5 is exactly ∆ + 1 and thus the total coloring conjecture holds for all pseudo- outerplanar graphs. 2 Main r esults and their pr oofs T o begin with, let us revie w an useful structural property of pseudo-outerplanar graphs which was prov ed in [6]. Lemma 1. Let G be a pseudo-outerplanar graph with minimum de gr ee at least two. Then (1) G has an edge uv such that d ( u ) = 2 and d ( v ) ≤ 4 , or (2) G has a 4 -cycle u xvy such that d ( u ) = d ( v ) = 2 , or (3) G has a 4 -cycle u xvy such that d ( u ) = d ( v ) = 3 and uv ∈ E ( G ) , or (4) G has a 7 -path x 0 u xvywy 0 such that d ( u ) = d ( v ) = d ( w ) = 2 , d ( x ) = d ( y ) = 5 and x x 0 , yy 0 , xy , x 0 y , xy 0 ∈ E ( G ) . Here it should be remarked that Lemma 1 is a straightforward simplification of the corresponding theorem in [6] (which had more subcases). Theorem 2. Let G be a pseudo-outerplanar graph with maximum degr ee ∆ and let M be an inte ger such that ∆ ≤ M . If M ≥ 5 , then χ 00 ( G ) ≤ M + 1 . Pr oof. W e shall pro ve the theorem by induction on | V ( G ) | + | E ( G ) | . So we as- sume that G is 2-connected and thus δ ( G ) ≥ 2. Set U = { 1 , · · · , M + 1 } . In the next, we complete the proof by verifying the following four claims; they imply a contradiction to Lemma 1. Claim 1 . If uv ∈ E ( G ) and d ( u ) = 2 , then d ( v ) ≥ M . Suppose, to the contrary , that d ( v ) ≤ M − 1. Consider the graph G 0 = G − uv . By induction, G 0 has an ( M + 1)-total coloring c . Denote the other neighbor of u by w . Now erase the color on u and color the edge uv with c ( uv ) ∈ A ( uv ) : = U \ ( { c ( v ) , c ( uw ) } ∪ { c ( v x ) | v x ∈ E ( G 0 ) } ). This is possible since | A ( uv ) | ≥ M + 1 − (2 + M − 2) = 1. Denote the extended coloring at this stage still by c . Then at last we color u with c ( u ) ∈ U \ { c ( v ) , c ( w ) , c ( uv ) , c ( uw ) } , which is also possible since | U | ≥ 6. Claim 2 . G does not contain a 4 -cycle u x vy such that d ( u ) = d ( v ) = 2 . 2 Otherwise, G 0 = G − { u , v } admits an ( M + 1)-total coloring by induction and ev ery edge of the 4-cycle has at least two av ailable colors since it is incident with at most ∆ − 1 colored elements. This implies that one can extend the coloring of G 0 to the four edges u x , v x , uy and vy since ev ery 4-cycle is 2-edge-choosable. At last, the two vertices u and v can be easily colored since they are both of degree two. Claim 3 . G does not contain a 4 -cycle u xvy such that d ( u ) = d ( v ) = 3 and uv ∈ E ( G ) . Suppose, to the contrary , that G contains such a 4-cycle. W e consider the graph G 0 = G − uv that has an ( M + 1)-total coloring c by induction. One can find that the only obstacle of extending c to uv is the case when M = 5 and U = { c ( u ) , c ( v ) , c ( u x ) , c ( uy ) , c ( v x ) , c ( vy ) } . W ithout loss of generality , let c ( u x ) = 1 , c ( uy ) = 2 , c ( v x ) = 3 , c ( vy ) = 4 , c ( u ) = 5 and c ( v ) = 6. If c ( x ) , 4, then we recolor u by 4 (note that c ( y ) , 4) and then color uv by 5. So we assume c ( x ) = 4. Similarly we can prov e c ( y ) = 3. Therefore, we can recolor v by 1 and then color uv by 6, a contradiction. Claim 4 . G does not contain a 7 -path x 0 u xvywy 0 such that d ( u ) = d ( v ) = d ( w ) = 2 , d ( x ) = d ( y ) = 5 and x x 0 , yy 0 , xy , x 0 y , xy 0 ∈ E ( G ) . Suppose, to the contrary , that G contains such a 7-path. W e consider the graph G 0 = G − { u , v , w } , which admits an ( M + 1)-total coloring c by induc- tion. Denote by A ( e ) the set of a v ailable colors to properly color an edge e ∈ { u x , v x , vy , wy , u x 0 , wy 0 } . It is easy to see that | A ( u x 0 ) | = | A ( wy 0 ) | = 1 and | A ( u x ) | = | A ( v x ) | = | A ( vy ) | = | A ( wy 0 ) | = 2, moreo ver , A ( u x ) = A ( v x ) and A ( vy ) = A ( wy ). If A ( u x 0 ) , A ( wy 0 ), without loss of generality assume that A ( u x 0 ) = { 1 } and A ( wy 0 ) = { 2 } , then color u x 0 with 1 and wy 0 with 2. If 1 ∈ A ( u x ), then color u x with c ( u x ) ∈ A ( u x ) \ { 1 } and v x with 1. If 1 < A ( u x ), then color v x with c ( v x ) ∈ A ( v x ) \ { 2 } and u x with c ( u x ) ∈ A ( u x ) \ { c ( v x ) } . In each case we have c ( v x ) , c ( wy 0 ). Thus we can color vy with c ( vy ) ∈ A ( vy ) \ { c ( v x ) } and wy with c ( wy ) ∈ A ( wy ) \ { c ( wy 0 ) } such that c ( vy ) , c ( wy ). At this stage, the three vertices u , v and w can be easily colored since they are all of degree two. So we assume that A ( u x 0 ) = A ( wy 0 ) = { 1 } . Now we firstly color u x 0 and wy 0 by 1. If 1 < A ( u x ), then color v x with c ( v x ) ∈ A ( v x ) and u x with c ( u x ) ∈ A ( u x ) \ { c ( v x ) } . The current extended coloring satisfies that c ( v x ) , c ( wy 0 ). Therefore, we can color the re- maining elements similarly as before. So we assume that 1 ∈ A ( u x ). This implies that 1 < { c ( x ) , c ( xy ) , c ( x x 0 ) , c ( xy 0 ) } and thus we can e xchange the colors on u x 0 and x x 0 . By doing so we obtain a new coloring satisfying c ( u x 0 ) , c ( wy 0 ) and therefore we can extend this partial coloring to G by a same argument as abov e.  Corollary 3. Every pseudo-outerplanar gr aph with maximum de gr ee ∆ ≥ 5 is ( ∆ + 1) -total colorable. Note that every graph with maximum degree ∆ ≤ 5 is ( ∆ + 2)-total colorable (see [2] and Chapter 4 of [5]). So we also hav e the following corollary . 3 Figure 1: A pseudo-outerplanar graph G with ∆ ( G ) = 3 and χ 00 ( G ) = 5 Corollary 4. The total coloring conjectur e holds for all pseudo-outerplanar graphs. The graph (a) in Figure 1 is a pseudo-outerplanar graph, since it has a pseudo- outerplanar drawing as (b). One can easy to check that it is a graph with maximum degree 3 and total chromatic number 5. Thus the upper bound for ∆ in Corollary 3, although probably not the best possible, cannot be less than 4. At last, we leave the following open problem to end this paper . Problem 5. T o determine the total chr omatic number of pseudo-outerplanar graphs with maximum degr ee four . Refer ences [1] M. Behzad. Graphs and their chromatic numbers. Doctoral thesis, Michigan State Univ ersity , 1965. [2] A. V . Kostochka. The total chromatic number of any multigraph with max- imum degree fiv e is at most seven. Discrete Mathematics. 162 (1996) 199– 214. [3] V . V izing. Some unsolved problems in graph theory . Uspekhi Mat. Nauk, 23 (1968) 117–134. [4] J. L. W u, D. Hu. T otal coloring of series-parallel graphs. Ars Comb . 73 (2004) 209–211. [5] H. P . Y ap. T otal colourings of graphs. Lecture Notes in Mathematics 1623, Berlin; London: Springer (1996). [6] X. Zhang, G. Liu, J. L. W u. Edge cov ering pseudo-outerplanar graphs with forests. arXiv:1108.3877v1 [math.CO]. [7] Z. Zhang, J. Zhang, J. W ang. The total chromatic number of some graphs. Scientia Sinica A 31 (1988) 1434–1441 4