The mean chromatic number of a graph is defined. This is a measure of the expected performance of the greedy vertex-colouring algorithm when each ordering of the vertices is equally likely. In this note, we analyse the asymptotic behaviour of the mean chromatic number for the paths and even cycles,
Induced Cycles and Chromatic Number
β Scribed by A.D. Scott
- Publisher
- Elsevier Science
- Year
- 1999
- Tongue
- English
- Weight
- 87 KB
- Volume
- 76
- Category
- Article
- ISSN
- 0095-8956
No coin nor oath required. For personal study only.
β¦ Synopsis
We prove that, for any pair of integers k, l 1, there exists an integer N(k, l ) such that every graph with chromatic number at least N(k, l ) contains either K k or an induced odd cycle of length at least 5 or an induced cycle of length at least l.
π SIMILAR VOLUMES
## Abstract In 1966 ErdΓΆs and Hajnal proved that the chromatic number of graphs whose odd cycles have lengths at most __l__ is at most __l__+1. Similarly, in 1992 GyΓ‘rfΓ‘s proved that the chromatic number of graphs which have at most __k__ odd cycle lengths is at most 2__k__+2 which was originally c
## Abstract For a graph __G__, let __g__(__G__) and Ο~g~(__G__) denote, respectively, the girth of __G__ and the number of cycles of length __g__(__G__) in __G__. In this paper, we first obtain an upper bound for Ο~g~(__G__) and determine the structure of a 2βconnected graph __G__ when Ο~g~(__G__)
Graph bundles generalize the notion of covering graphs and products of graphs. The chromatic numbers of product bundles with respect to the Cartesian, strong and tensor product whose base and fiber are cycles are determined. ## 1. Introduction If G is a graph, V(G) and E(G) denote its vertex and e