On ${\bi d}$ -Diagonal Colorings of Embedded Graphs of Low Maximum Face Size

Abstract

The problems of cyclic colorings (due to Ore and Plummer) and diagonal colorings (due to Bouchet, Fouquet, Jolivet, and Riviere) were simultaneously generalized by Hornak and Jendrol into d-diagonal colorings. A coloring of a graph embedded on a surface is d-diagonal if any pair of vertices which are in the same face after the deletion of at most d edges of the graph are colored differently. A cyclic coloring is a 0-diagonal coloring, and a diagonal coloring is a 1-diagonal coloring. These types of problems have proved hard for graphs of low maximum face size. Equivalent problems are the Four Color Theorem, Ringel's 1-embeddable problem, and the Nine Color Conjecture. Previously, general results for d-diagonal colorings have only been established for maximum face size at least eight, and for graphs all of whose faces are the same size. This paper establishes the first results on d-diagonal colorings for graphs whose maximum face size is between four and seven, and improves the known upper bounds for maximum face size between eight and ten. The proofs use five instances of the Discharging Method.