Weakening Total Coloring Conjecture: Weak TCC and Hadwiger's Conjecture on Total Graphs

Hadwiger's conjecture is one of the most important and long-standing conjectures in graph theory. Reed and Seymour showed in 2004 that Hadwiger's conjecture is true for line graphs. We investigate this conjecture on the closely related class of total graphs. The total graph of $G$, denoted by $T(G)$, is defined on the vertex set $V(G)\sqcup E(G)$ with $c_1,c_2\in V(G)\sqcup E(G)$ adjacent whenever $c_1$ and $c_2$ are adjacent to or incident on each other in $G$. We first show that there exists a constant $C$ such that, if the connectivity of $G$ is at least $C$, then Hadwiger's conjecture is true for $T(G)$. The total chromatic number $\chi"(G)$ of a graph $G$ is defined to be equal to the chromatic number of its total graph. That is, $\chi"(G)=\chi(T(G))$. Another well-known conjecture in graph theory, the total coloring conjecture or TCC, states that for every graph $G$, $\chi"(G)\leq\Delta(G)+2$, where $\Delta(G)$ is the maximum degree of $G$. We show that if a weaker version of the total coloring conjecture (weak TCC) namely, $\chi"(G)\leq\Delta(G)+3$, is true for a class of graphs $\mathcal{F}$ that is closed under the operation of taking subgraphs, then Hadwiger's conjecture is true for the class of total graphs of graphs in $\mathcal{F}$. This motivated us to look for classes of graphs that satisfy weak TCC. It may be noted that a complete proof of TCC for even 4-colorable graphs (in fact even for planar graphs) has remained elusive even after decades of effort; but weak TCC can be proved easily for 4-colorable graphs. We noticed that in spite of the interest in studying $\chi"(G)$ in terms of $\chi(G)$ right from the initial days, weak TCC is not proven to be true for $k$-colorable graphs even for $k=5$. In the second half of the paper, we make a contribution to the literature on total coloring by proving that $\chi"(G)\leq\Delta(G)+3$ for every 5-colorable graph $G$.