Unique colorability and clique minors

TL;DR

This paper investigates the properties of graphs with a unique coloring, proving new bounds on clique minors and subgraph structures for such graphs, especially when the chromatic number is at most 6 or the graph has no antitriangles.

Contribution

It generalizes Hadwiger's and Seymour's conjectures for graphs with exactly one c(G)-coloring, providing bounds without relying on the four-color theorem.

Findings

If c(G) ≤ 6 and G has exactly one c(G)-coloring, then h(G) ≥ c(G).

If G has no antitriangles and exactly one c(G)-coloring, then s(G) ≥ |V(G)|/2.

Abstract

For a graph G, let h(G) denote the largest k such that G has k pairwise disjoint pairwise adjacent connected nonempty subgraphs, and let s(G) denote the largest k such that G has k pairwise disjoint pairwise adjacent connected subgraphs of size 1 or 2. Hadwiger's conjecture states that h(G) is at least c(G), where c(G) is the chromatic number of G. Seymour conjectured that s(G) is at least |V(G)|/2 for all graphs without antitriangles, i. e. three pairwise nonadjacent vertices. Here we concentrate on graphs G with exactly one c(G)-coloring. We prove generalizations of (i) if c(G) is at most 6 and G has exactly one c(G)-coloring then h(G) is at least c(G), where the proof does not use the four-color-theorem, and (ii) if G has no antitriangles and G has exactly one c(G)-coloring then s(G) is at least |V(G)|/2.