Arc coloring of odd graphs for hamiltonicity

TL;DR

This paper explores arc colorings in odd graphs, demonstrating a specific coloring scheme that influences Hamiltonian cycle constructions and factorization properties in these graphs and their related structures.

Contribution

It introduces a novel arc coloring method for odd graphs that impacts Hamiltonicity and factorization analysis in these graphs and their double coverings.

Findings

Established an arc factorization with colors 0 to k in odd graphs

Analyzed the impact of arc colorings on Hamilton cycles in odd graphs

Connected arc colorings to properties of middle-levels graphs

Abstract

Coloring the arcs of biregular graphs was introduced with possible applications to industrial chemistry, molecular biology, cellular neuroscience, etc. Here, we deal with arc coloring in some non-bipartite graphs. In fact, for $1<k\in\mathbb{Z}$, we find that the odd graph $O_k$ has an arc factorization with colors $0,1,\ldots,k$ such that the sum of colors of the two arcs of each edge equals $k$. This is applied to analyzing the influence of such arc factorizations in recently constructed uniform 2-factors in $O_k$ and in Hamilton cycles in $O_k$ as well as in its double covering graph known as the middle-levels graph $M_k$.