Strong parity edge-colorings of graphs

TL;DR

This paper characterizes strong parity edge-colorings of graphs, establishes bounds, and solves several open problems, including exact values for complete bipartite graphs and hypercube subgraphs, while disproving a conjecture about bipartite graphs.

Contribution

It provides a characterization of strong parity edge-colorings, proves the conjecture for complete bipartite graphs, and disproves a related conjecture for bipartite graphs, advancing understanding of these colorings.

Findings

Proved the conjecture that ap; p(G) for complete bipartite graphs equals the Hopf-Stiefel function.

Established that ap; p(G) equals eil; log_2 n eil; for certain graphs, specifically subgraphs of hypercubes.

Asymptotically computed ap; p(G) for the ll;th distance-power of a path, showing ap; p(P_n^ll;) pprox; ll; eil; log_2 n eil;.

Abstract

An edge-coloring of a graph $G$ assigns a color to each edge of $G$. An edge-coloring is a parity edge-coloring if for each path $P$ in $G$, it uses some color on an odd number of edges in $P$. It is a strong parity edge-coloring if for every open walk $W$ in $G$, it uses some color an odd number of times along $W$. The minimum numbers of colors in parity and strong parity edge-colorings of $G$ are denoted $p(G)$ and $\hat{p}(G)$, respectively. We characterize strong parity edge-colorings and use this characterization to prove lower bounds on $\hat{p}(G)$ and answer several questions of Bunde, Milans, West, and Wu. The applications are as follows. (1) We prove the conjecture that $\hat{p}(K_{s,t})=s \circ t$, where $s \circ t$ is the Hopf-Stiefel function. (2) We show that $\hat{p}(G)$ for a connected $n$-vertex graph $G$ equals the known lower bound $\lceil \log_2 n \rceil$ if and only if $G$ is a subgraph of the hypercube $Q_{\lceil \log_2 n \rceil }$. (3) We asymptotically compute $\hat{p}(G)$ when $G$ is the $\ell$th distance-power of a path, proving $\hat{p}(P_n^\ell)\sim\ell \lceil {\log_2 n} \rceil$. (4) We disprove the conjecture that $\hat{p}(G)=p(G)$ when $G$ is bipartite by constructing bipartite graphs $G$ such that $\hat{p}(G)/p(G)$ is arbitrarily large; in particular, with $\hat{p}(G)\ge\frac{1-o(1)}3 k\ln k$ and $p(G)\le2k+k^{1/3}$.