Non‐rainbow colorings of 3‐, 4‐ and 5‐connected plane graphs

Abstract

We study vertex-colorings of plane graphs that do not contain a rainbow face, i.e., a face with vertices of mutually distinct colors. If G is a 3-connected plane graph with n vertices, then the number of colors in such a coloring does not exceed 7n−8 9. If G is 4-connected, then the number of colors is at most 5n−6 8 , and for n ≡ 3 (mod 8), it is at most 5n−6 8 − 1. Finally, if G is 5-connected, then the number of colors is at most 25 58 n − 22 29 . The bounds for 3-connected and 4-connected plane graphs are the best possible as we exhibit constructions of graphs with colorings matching the bounds. c

Null and non-rainbow colorings of projective plane and sphere triangulations

“…Finally, in Section 4 we provide as a corollary of Theorem 1, the exact value from which tricolored faces are inevitable in any coloring of a given sphere triangulation, respectively in any coloring of a given projective plane triangulation. The problem of maximizing the number of colors in a vertex coloring avoiding rainbow faces has been studied recently in several papers [3,5,6,8,9]. In Section 4, before stating and proving our result, we summarize some known results in the area. )…”

Rainbow faces in edge‐colored plane graphs

“…Negami [19] investigated plane triangulations and showed that f (G) ≤ 2 (G). Dvořák et al in [8] proved that for every 3-connected plane graph G of order n it holds that f (G) ≤ (7n −8)/9 , for every 4-connected graph G it holds that f (G) ≤ (5n −6)/8 if n ≡ 3 (mod 8), and f (G) ≤ (5n −6)/8 +1 if n ≡ 3 (mod 8), and for every 5-connected plane graph G f (G) ≤ Besides results on non-rainbow colorings of plane graphs with no short cycles and non-trivially connected plane graphs, there are also results on specific families of plane graphs, e.g. the numbers f (G) were also determined for semiregular polyhedra by Jendrol' and Schrötter [14].…”

Evolutionary design of non-frustrated networks of phase-repulsive oscillators

“…As we show, in opposition to the well studied synchronisable networks35, whose synchronisation performance is typically enhanced by hierarchical or scale-free structures37, non-frustrated networks are topologically very uniform, and exhibit a precise scale3839 rather than being scale-free. This problem is similar to the problem of 2-color vertex colouring encountered in graph theory4041. The difference is that the process of vertex colouring is here automatically done by the network phase-repulsive dynamics.…”