Edge-choosability and total-choosability of planar graphs with no adjacent 3-cycles

Daniel W. Cranston

Discussiones Mathematicae Graph Theory, Tome 29 (2009), p. 163-178 / Harvested from The Polish Digital Mathematics Library

Résumé

Let G be a planar graph with no two 3-cycles sharing an edge. We show that if Δ(G) ≥ 9, then χ'ₗ(G) = Δ(G) and χ''ₗ(G) = Δ(G)+1. We also show that if Δ(G) ≥ 6, then χ'ₗ(G) ≤ Δ(G)+1 and if Δ(G) ≥ 7, then χ''ₗ(G) ≤ Δ(G)+2. All of these results extend to graphs in the projective plane and when Δ(G) ≥ 7 the results also extend to graphs in the torus and Klein bottle. This second edge-choosability result improves on work of Wang and Lih and of Zhang and Wu. All of our results use the discharging method to prove structural lemmas about the existence of subgraphs with small degree-sum. For example, we prove that if G is a planar graph with no two 3-cycles sharing an edge and with Δ(G) ≥ 7, then G has an edge uv with d(u) ≤ 4 and d(u)+d(v) ≤ Δ(G)+2. All of our proofs yield linear-time algorithms that produce the desired colorings.

Détail

Publié le : 2009-01-01

Zbl 1198.05044

EUDML-ID : urn:eudml:doc:270180

Bibliographie

[000] [1] M. Behzad, Graphs and their chromatic numbers (Ph.D. thesis, Michigan State University, 1965).

[001] [2] B. Bollabás and A.J. Harris, List-colorings of graphs, Graphs Combin. 1 (1985) 115-127, doi: 10.1007/BF02582936.

[002] [3] O.V. Borodin, Generalization of a theorem of Kotzig and a prescribed coloring of the edges of planar graphs, Mathematical Notes of the Academy of Sciences of the USSR 48 (1990) 1186-1190, doi: 10.1007/BF01240258. | Zbl 0742.05039

[003] [4] O.V. Borodin, On the total coloring of planar graphs, J. reine angew. Math. 394 (1989) 180-185, doi: 10.1515/crll.1989.394.180. | Zbl 0653.05029

[004] [5] O.V. Borodin, Structural properties of plane graphs without adjacent triangles and an application to 3-colorings, J. Graph Theory 21 (1996) 183-186, doi: 10.1002/(SICI)1097-0118(199602)21:2<183::AID-JGT7>3.0.CO;2-N | Zbl 0838.05039

[005] [6] O.V. Borodin, A.V. Kostochka and D.R. Woodall, List edge and list total colourings of multigraphs, J. Comb. Theory (B) 71 (1997) 184-204, doi: 10.1006/jctb.1997.1780. | Zbl 0876.05032

[006] [7] P. Erdös, A.L. Rubin and H. Taylor, Choosability in graphs, Proceedings of the West Coast Conference on Combinatorics, Graph Theory and Computing (Arcata, California, 1979), Congressus Numeratium 26 (1980) 125-157.

[007] [8] R.P. Gupta, The chromatic index and the degree of a graph (Abstract 66T-429), Notices Amer. Math. Soc. 13 (1966) 719.

[008] [9] T.R. Jensen and B. Toft, Graph Coloring Problems (John Wiley & Sons, 1995).

[009] [10] M. Juvan, B. Mohar and R. Skrekovski, Graphs of degree 4 are 5-edge-choosable, J. Graph Theory, 32 (1999) 250-264, doi: 10.1002/(SICI)1097-0118(199911)32:3<250::AID-JGT5>3.0.CO;2-R | Zbl 0934.05052

[010] [11] A.V. Kostochka, List edge chromatic number of graphs with large girth, Discrete Math. 101 (1992) 189-201, doi: 10.1016/0012-365X(92)90602-C. | Zbl 0766.05027

[011] [12] A.V. Kostochka, The total coloring of a multigraph with maximal degree 4, Discrete Math. 17 (1977) 161-163, doi: 10.1016/0012-365X(77)90146-7. | Zbl 0411.05038

[012] [13] A.V. Kostochka, The total chromatic number of a multigraph with maximal degree 5 is at most 7, Discrete Math. 162 (1996) 199-214, doi: 10.1016/0012-365X(95)00286-6. | Zbl 0871.05023

[013] [14] A.V. Kostochka, Exact upper bound for the total chromatic number of a graph, Proc. 24th International Wiss. Koll., Tech. Hochsch. Ilmenau, (1979) 33-36 (in Russian).

[014] [15] M. Rosenfeld, On the total coloring of certain graphs, Israel J. Math. 9 (1971) 396-402, doi: 10.1007/BF02771690. | Zbl 0211.56604

[015] [16] N. Vijayaditya, On total chromatic number of a graph, J. London Math. Soc. 3 (1971) 405-408, doi: 10.1112/jlms/s2-3.3.405. | Zbl 0223.05103

[016] [17] V.G. Vizing, On an estimate of the chromatic class of a p-graph, Diskret. Analiz. 3 (1964) 25-30 (in Russian).

[017] [18] V.G. Vizing, Coloring the vertices of a graph in prescribed colors, Diskret. Analiz., No. 29 Metody Diskret. Anal. v Teorii Kodov i Shem (1976), 3-10, 101 (in Russian).

[018] [19] V.G. Vizing, Critical graphs with a given chromatic class., Diskret. Analiz. 5 (1965) 9-17 (in Russian).

[019] [20] W. Wang and K. Lih, Choosability and edge choosability of planar graphs without intersecting triangles, SIAM J. Discrete Math. 15 (2002) 538-545, doi: 10.1137/S0895480100376253. | Zbl 1006.05024

[020] [21] H.P. Yap, Total-colourings of graphs, Manuscript, 1989. | Zbl 0636.05025

[021] [22] L. Zhang and B. Wu, Edge choosability of planar graphs without small cycles, Discrete Math. 283 (2004) 289-293, doi: 10.1016/j.disc.2004.01.001. | Zbl 1042.05033