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

Abstract

The total graph of a graph $G$, denoted by $T(G)$, is defined on the vertex set $V(G)\cup E(G)$ with $c_1,c_2 \in V(G)\cup E(G)$ adjacent whenever $c_1$ and $c_2$ are adjacent to (or incident on) each other in $G$. The total chromatic number $\chi''(G)$ 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 $\Delta(G)$, $\chi''(G)\leq \Delta(G) + 2$. In this paper, we consider two ways to weaken TCC: (1)  Weak TCC:  This conjecture states that for a simple finite graph $G$, $\chi''(G) = \chi(T(G)) \leq\Delta(G) + 3$. 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 $\chi$-bounding function for the class of total graphs in the following way: If $H$ is the total graph of a simple finite graph, then $\chi(H) \leq\omega(H) + 1$, where $\omega(H)$ is the clique number of $H$. A natural way to relax this question is to replace $\omega(H)$ by the Hadwiger number $\eta(H)$, 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 $\chi(H) \leq \eta(H)$. We prove that this is true if $H$ is the total graph of a graph with sufficiently large connectivity. A second motivation for studying Hadwiger's conjecture for total graphs is the following: Consider the class of split graphs whose vertex set is partitioned into an independent set $A$ and a clique $B$, with the following additional constraints: (1) Each vertex in $B$ has exactly 2 neighbours in $A$; (2) No two vertices in $B$ have the same neighbourhood in $A$. It is known that (European Journal of Combinatorics, 76, 159-174,2019)  if Hadwiger's conjecture is proved for the squares of this special class of split graphs, then it holds also for the general case. Of course, proving the conjecture for this specialzed-looking case is indeed difficult since it is only a reformulation of the general case, and therefore it is natural to consider the difficulty level of Hadwiger's conjecture for the squares of graph classes defined by slighly modifying the above class of graphs. A natural structural modification is to assume that both $A$ and $B$ are independent sets, keeping everything else same. It turns out that the squares of this modified class of graphs is exactly the class of total graphs.  From this perspective, it is not really surprising that HC on Total Graphs is also challenging. On the other hand, we show that weak TCC implies HC on total graphs. This perhaps suggests that the latter is an easier problem than the former.