Properties, Proved and Conjectured, of Keller, Mycielski, and Queen Graphs

TL;DR

This paper investigates properties of Keller, Mycielski, and Queen graphs, proving several results including edge-coloring conjectures, Hamiltonian connectivity, and exact graph invariants, supported by computational methods and new algorithms.

Contribution

It provides new proofs, conjectures, and computational algorithms for key properties of these graph families, advancing understanding of their chromatic and Hamiltonian characteristics.

Findings

Edge-chromatic number of queen graphs mostly determined, conjecture supported by computation.

Mycielski graphs are Hamilton-connected, except for M(3).

Exact values of chromatic number, edge-chromatic number, and independence number for Keller graphs.

Abstract

We prove several results about three families of graphs. For queen graphs, defined from the usual moves of a chess queen, we find the edge-chromatic number in almost all cases. In the unproved case, we have a conjecture supported by a vast amount of computation, which involved the development of a new edge-coloring algorithm. The conjecture is that the edge-chromatic number is the maximum degree, except when simple arithmetic forces the edge-chromatic number to be one greater than the maximum degree. For Mycielski graphs, we strengthen an old result that the graphs are Hamiltonian by showing that they are Hamilton-connected (except M(3), which is a cycle). For Keller graphs G(d), we establish, in all cases, the exact value of the chromatic number, the edge-chromatic number, and the independence number, and we get the clique covering number in all cases except 5 <= d <= 7. We also investigate Hamiltonian decompositions of Keller graphs, obtaining them up to G(6).