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

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

#### 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.

Download#### 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