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## Discussiones Mathematicae Graph Theory

2009 | 29 | 1 | 163-178
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
163-178
Opis fizyczny
Daty
wydano
2009
otrzymano
2008-01-18
poprawiono
2008-10-20
zaakceptowano
2008-10-20
Twórcy
autor
• University of Illinois, Urbana, USA
Bibliografia
• [1] M. Behzad, Graphs and their chromatic numbers (Ph.D. thesis, Michigan State University, 1965).
• [2] B. Bollabás and A.J. Harris, List-colorings of graphs, Graphs Combin. 1 (1985) 115-127, doi: 10.1007/BF02582936.
• [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.
• [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.
• [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
• [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.
• [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.
• [8] R.P. Gupta, The chromatic index and the degree of a graph (Abstract 66T-429), Notices Amer. Math. Soc. 13 (1966) 719.
• [9] T.R. Jensen and B. Toft, Graph Coloring Problems (John Wiley & Sons, 1995).
• [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
• [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.
• [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.
• [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.
• [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).
• [15] M. Rosenfeld, On the total coloring of certain graphs, Israel J. Math. 9 (1971) 396-402, doi: 10.1007/BF02771690.
• [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.
• [17] V.G. Vizing, On an estimate of the chromatic class of a p-graph, Diskret. Analiz. 3 (1964) 25-30 (in Russian).
• [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).
• [19] V.G. Vizing, Critical graphs with a given chromatic class., Diskret. Analiz. 5 (1965) 9-17 (in Russian).
• [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.
• [21] H.P. Yap, Total-colourings of graphs, Manuscript, 1989.
• [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.
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Bibliografia
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