Upper Bounds for the Total Chromatic Number of Join Graphs and Cobipartite Graphs

Leandro M. Zatesko, Renato Carmo, André L. P. Guedes

2018

Abstract

We concern ourselves with the combinatorial optimisation problem of determining a minimum total colouring of a graph G for the case wherein G is a join graph G = G_1 ∗ G_2 or a cobipartite graph G = (V_1 ∪ V_2, E(G)). We present algorithms for computing a feasible, not necessarily optimal, solution for this problem, providing the following upper bounds for the total chromatic numbers of these graphs (let n_i := |V_i| and _i := (G_i) for i ∈ {1, 2} and  ∈ {∆, χ, χ′, χ′′}): χ′′(G) ≤ max{n_1, n_2} + 1 + P(G_1, G_2) if G is a join graph, wherein P(G_1, G_2) := min{∆_1 + ∆_2 + 1, max{χ′_1, χ′′_2}}; χ′′(G) ≤ max{n_1, n_2} + 2(max{∆^B_1, ∆^B_2} + 1) if G is cobipartite, wherein ∆^B_i := max_{u ∈ V_i} d_{G[∂_G(V_i)]}(u) for i ∈ {1, 2}. Our algorithm for the cobipartite graphs runs in polynomial time. Our algorithm for the join graphs runs in polynomial time if P(G_1, G_2) is replaced by ∆_1 + ∆_2 + 1 or if it can be computed in polynomial time. We also prove the Total Colouring Conjecture for some subclasses of join graphs, such as some joins of indifference (unitary interval) graphs.

Paper Citation

in Harvard Style

Zatesko L., Carmo R. and Guedes A. (2018). Upper Bounds for the Total Chromatic Number of Join Graphs and Cobipartite Graphs.In Proceedings of the 7th International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES, ISBN 978-989-758-285-1, pages 247-253. DOI: 10.5220/0006627102470253

in Bibtex Style

@conference{icores18,
author={Leandro M. Zatesko and Renato Carmo and André L. P. Guedes},
title={Upper Bounds for the Total Chromatic Number of Join Graphs and Cobipartite Graphs},
booktitle={Proceedings of the 7th International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,},
year={2018},
pages={247-253},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0006627102470253},
isbn={978-989-758-285-1},
}

in EndNote Style

TY - CONF

JO - Proceedings of the 7th International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,
TI - Upper Bounds for the Total Chromatic Number of Join Graphs and Cobipartite Graphs
SN - 978-989-758-285-1
AU - Zatesko L.
AU - Carmo R.
AU - Guedes A.
PY - 2018
SP - 247
EP - 253
DO - 10.5220/0006627102470253