Weak Total Coloring Conjecture and Hadwiger's Conjecture on Total Graphs

The total graph of a graph G, denoted by T(G), is defined on the vertex set V (G) boolean OR E(G) with C1, C2 is an element of V (G) boolean OR E(G) adjacent whenever C1 and C2 are adjacent to (or incident on) each other in G. The total chromatic number x ''(G) of a graph G is defined to be the chromatic number of its total graph. The well-known Total Coloring Conjecture or TCC states that for every simple finite graph G having maximum degree increment (G), x ''(G) increment (G) + 2. In this paper, we consider two ways to weaken TCC:1. Weak TCC: This conjecture states that for a simple finite graph G, x ''(G) = x(T(G)) increment (G)+3. While weak TCC is known to be true for 4-colorable graphs, it has remained open for 5-colorable graphs. In this paper, we settle this long pending case.2. Hadwiger's Conjecture for total graphs: We can restate TCC as a conjecture that proposes the existence of a strong x-bounding function for the class of total graphs in the following way: If H is the total graph of a simple finite graph, then x(H) w(H) + 1, where w(H) is the clique number of H. A natural way to relax this question is to replace w(H) by the Hadwiger number n(H), the number of vertices in the largest clique minor of H. This leads to the Hadwiger's Conjecture (HC) for total graphs: if H is a total graph then x(H) n(H). We prove that this is true if H is the total graph of a graph with sufficiently large connectivity. It is known that (European Journal of Combinatorics, 76, 159-174,2019) if Hadwiger's Conjecture is proved for the squares of certain special class of split graphs, then it holds also for the general case. The class of total graphs turns out to be the squares of graphs obtained by a natural structural modification of this type of split graphs.