| 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. |