On triangle-free graphs of order 10 with prescribed 1-defective chromatic number
Combinatorics
2015-01-20 v1
Abstract
A graph is (m, k)-colourable if its vertices can be coloured with m colours such that the maximum degree of any subgraph induced on ver- tices receiving the same colour is at most k. The k-defective chromatic number for a graph is the least positive integer m for which the graph is (m, k)-colourable. All triangle-free graphs on 8 or fewer vertices are (2, 1)-colourable. There are exactly four triangle-free graphs of order 9 which have 1-defective chromatic number 3. We show that these four graphs appear as subgraphs in almost all triangle-free graphs of order 10 with 1-defective chromatic number equal to 3. In fact there is a unique triangle-free (3, 1)-critical graph on 10 vertices and we exhibit this graph.
Keywords
Cite
@article{arxiv.1501.04384,
title = {On triangle-free graphs of order 10 with prescribed 1-defective chromatic number},
author = {Nirmala Achuthan and N. R. Achuthan and G. Keady},
journal= {arXiv preprint arXiv:1501.04384},
year = {2015}
}
Comments
17pages, 5figures