A new upper bound for the chromatic number of a graph

Ingo Schiermeyer

ArticleOriginal scientific text

Title

A new upper bound for the chromatic number of a graph

Authors ¹

Affiliations

Institut für Diskrete Mathematik und Algebra, Technische Universität Bergakademie Freiberg, 09596 Freiberg, Germany

Abstract

Let G be a graph of order n with clique number ω(G), chromatic number χ(G) and independence number α(G). We show that χ(G) ≤ [(n+ω+1-α)/2]. Moreover, χ(G) ≤ [(n+ω-α)/2], if either ω + α = n + 1 and G is not a split graph or α + ω = n - 1 and G contains no induced

K_{ω + 3} - C ₅

.

Keywords

ENG

Vertex colouring, chromatic number, upper bound

B. Baetz and D.R. Wood, Brooks' Vertex Colouring Theorem in Linear Time, TR CS-AAG-2001-05, Basser Dep. Comput. Sci., Univ. Sydney, (2001) 4 pages.
C. Berge, Les problèms de coloration en théorie des graphes, Publ. Inst. Statist. Univ. Paris 9 (1960) 123-160.
C. Berge, Perfect graphs, in: Six papers on graph theory, Indian Statistical Institute, Calcutta (1963), 1-21.
R.C. Brigham and R.D. Dutton, A Compilation of Relations between Graph Invariants, Networks 15 (1985) 73-107, doi: 10.1002/net.3230150108.
R.L. Brooks, On colouring the nodes of a network, Proc. Cambridge Phil. Soc. 37 (1941) 194-197, doi: 10.1017/S030500410002168X.
M. Chudnovsky, N. Robertson, P. Seymour and R. Thomas, Progress on perfect graphs, Math. Program. (B) 97 (2003) 405-422.
M. Chudnovsky, N. Robertson, P. Seymour and R. Thomas, The strong perfect graph theorem, Ann. Math. (2) 164 (2006) 51-229, doi: 10.4007/annals.2006.164.51.
M. Chudnovsky, G. Cornuéjols, X. Liu, P. Seymour and K. Vusković, Recognizing Berge Graphs, Combinatorica 25 (2005) 143-186, doi: 10.1007/s00493-005-0012-8.
P. Erdös, Graph theory and probability, Canad. J. Math. 11 (1959) 34-38, doi: 10.4153/CJM-1959-003-9.
J.L. Gross and J. Yellen, Handbook of Graph Theory (CRC Press, 2004).
L. Lovász, Three short proofs in graph theory, J. Combin. Theory (B) 19 (1975) 269-271, doi: 10.1016/0095-8956(75)90089-1.
E.A. Nordhaus and J.W. Gaddum, On complementary graphs, Amer. Math. Monthly 63 (1956) 175-177, doi: 10.2307/2306658.
B. Randerath and I. Schiermeyer, Vertex colouring and forbidden subgraphs - a survey, Graphs and Combinatorics 20 (2004) 1-40, doi: 10.1007/s00373-003-0540-1.

Title

A new upper bound for the chromatic number of a graph

Affiliations

Abstract

Keywords

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