Well-hued graphs with first difference two

TL;DR

This paper studies well-hued graphs, characterizes their sequences, proves a conjecture about their uniqueness, and explores conditions when both a graph and its complement are well-hued.

Contribution

It proves a conjecture on the uniqueness of connected well-hued graphs with specific parameters and characterizes nearly all well-hued graphs with a given initial sequence value.

Findings

Proved the conjecture on the uniqueness of certain well-hued graphs.

Characterized nearly all well-hued graphs with initial sequence value 2.

Investigated conditions for both a graph and its complement to be well-hued.

Abstract

A graph $G$ is said to be well-hued if every maximal $k$-colorable subgraph of $G$ has the same order $a_k$. Therefore, if $G$ is well-hued, we can associate with $G$ a sequence $\{a_k\}$. Necessary and sufficient conditions were given as to when a sequence $\{a_k\}$ is realized by a well-hued graph. Further, it was conjectured there is only one connected well-hued graph with $a_2 = a_1 + 2$ for every $a_1 \ge 4$. In this paper, we prove this conjecture as well as characterize nearly all well-hued graphs with $a_1=2$. We also investigate when both $G$ and its complement are well-hued.