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arXiv · 2503.20055

From semi-total to equitable total colorings, and their induced partitions into perfect and total perfect codes

I. J. Dejter
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Abstract

Independently posed by Behzad and Vizing, the Total Coloring Conjecture asserts that the total chromatic number of a connected simple graph $G$ is either $\Delta(G)+1$ or $\Delta(G)+2$, where $\Delta(G)$ is the largest degree of a vertex of $G$. Deciding whether a cubic graph $G$ has total chromatic number $\Delta(G)+1$, even for bipartite cubic graphs, is NP-hard. The problem persists even for equitable total colorings. Williams and Holroyd gave a condition for total colorings via the introduction of semi-total colorings. In the present work, focus is set on obtaining, by means of a variation of Kempe's 1879 graph-coloring algorithm, equitable total colorings out of semi-total colorings of: symmetric cubic graphs, cage graphs and expansions of cubic graphs. It is noted that total colorings of the 3-cube and subsequent polygon prisms exist having perfect codes as their vertex color classes, as well as semi-total colorings of M\"obius ladders having total perfect codes as their vertex color classes, only one of which, in $K_{3,3}$, being equitable; in addition, a partition into two perfect-codes and one total-perfect code is given by the vertex color classes of the 90-vertex Foster graph. On the other hand, a semi-total coloring of the 42-vertex $(5,6)$-cage yields its decomposition into three copies of the Heawood graph via its edges of same color endvertices and seven remaining 6-cycles.

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I. J. Dejter. 2025-03-25. From semi-total to equitable total colorings, and their induced partitions into perfect and total perfect codes. https://arxiv.org/abs/2503.20055

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