PDF
Discussiones Mathematicae Graph Theory 27(1) (2007)
137-142
DOI: https://doi.org/10.7151/dmgt.1350
A NEW UPPER BOUND FOR THE CHROMATIC NUMBER OF A GRAPH
Ingo Schiermeyer
Institut für Diskrete Mathematik und Algebra
Technische Universität Bergakademie Freiberg
09596 Freiberg, Germany
e-mail: schierme@tu-freiberg.de
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 Kω+3− C5.Keywords: Vertex colouring, chromatic number, upper bound.
2000 Mathematics Subject Classification: 05C15, 05C69.
References
| [1] | 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. |
| [2] | C. Berge, Les problèms de coloration en théorie des graphes, Publ. Inst. Statist. Univ. Paris 9 (1960) 123-160. |
| [3] | C. Berge, Perfect graphs, in: Six papers on graph theory, Indian Statistical Institute, Calcutta (1963), 1-21. |
| [4] | R.C. Brigham and R.D. Dutton, A Compilation of Relations between Graph Invariants, Networks 15 (1985) 73-107, doi: 10.1002/net.3230150108. |
| [5] | R.L. Brooks, On colouring the nodes of a network, Proc. Cambridge Phil. Soc. 37 (1941) 194-197, doi: 10.1017/S030500410002168X. |
| [6] | M. Chudnovsky, N. Robertson, P. Seymour and R. Thomas, Progress on perfect graphs, Math. Program. (B) 97 (2003) 405-422. |
| [7] | 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. |
| [8] | 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. |
| [9] | P. Erdös, Graph theory and probability, Canad. J. Math. 11 (1959) 34-38, doi: 10.4153/CJM-1959-003-9. |
| [10] | J.L. Gross and J. Yellen, Handbook of Graph Theory (CRC Press, 2004). |
| [11] | L. Lovász, Three short proofs in graph theory, J. Combin. Theory (B) 19 (1975) 269-271, doi: 10.1016/0095-8956(75)90089-1. |
| [12] | E.A. Nordhaus and J.W. Gaddum, On complementary graphs, Amer. Math. Monthly 63 (1956) 175-177, doi: 10.2307/2306658. |
| [13] | 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. |
Received 20 January 2006
Revised 28 December 2006
Close