On a Sufficient Condition for Planar Graphs of Maximum Degree 6 to be Totally 7-Colorable

A total $k$-coloring of a graph is an assignment of $k$ colors to its vertices and edges such that no two adjacent or incident elements receive the same color. The Total Coloring Conjecture (TCC) states that every simple graph $G$ has a total ($\Delta(G)+2$)-coloring, where $\Delta(G)$ is the maximum degree of $G$. This conjecture has been confirmed for planar graphs with maximum degree at least 7 or at most 5, i.e., the only open case of TCC is that of maximum degree 6. It is known that every planar graph $G$ of $\Delta(G) \geq 9$ or $\Delta(G) \in \{7, 8\}$ with some restrictions has a total $(\Delta(G) + 1)$-coloring. In particular, in [Shen and Wang, "On the 7 total colorability of planar graphs with maximum degree 6 and without 4-cycles", Graphs and Combinatorics, 25: 401-407, 2009], the authors proved that every planar graph with maximum degree 6 and without 4-cycles has a total 7-coloring. In this paper, we improve this result by showing that every diamond-free and house-free planar graph of maximum degree 6 is totally 7-colorable if every 6-vertex is not incident with two adjacent 4-cycles or not incident with three cycles of size $p,q,\ell$ for some $\{p,q,\ell\}\in \{\{3,4,4\},\{3,3,4\}\}$.