Equitable factorizations of highly edge-connected graphs: complete characterizations

TL;DR

This paper provides complete characterizations for equitable and almost equitable factorizations of highly edge-connected graphs, establishing new criteria and conditions for such decompositions with applications to parity and degree constraints.

Contribution

It introduces novel necessary and sufficient conditions for equitable factorizations in highly edge-connected graphs, including simplified criteria and applications to degree and parity constrained decompositions.

Findings

Characterization of equitable factorizations under degree conditions

Simplified criteria for degree-based factorizations

Applications to parity and almost even factorizations

Abstract

In this paper, we show that every highly edge-connected graph $G$, under a necessary and sufficient degree condition, can be edge-decomposed into $k$ factors $G_1,\ldots, G_k$ such that for each vertex $v\in V(G_i)$ with $1\le i\le k$, $|d_{G_i}(v)-d_G(v)/k|<1$. This characterization covers graphs having at least $k-1$ vertices with degree not divisible by $k$. In addition, we investigate almost equitable factorizations in arbitrary edge-connected graphs. Next, we establish a simpler criterion for the existence of factorizations $G_1,\ldots, G_k$ satisfying $d_{G_i}(v)\ge \lfloor d_G(v)/k\rfloor$ for all vertices $v$ (reps. $d_{G_i}(v)\le \lceil d_G(v)/k\rceil$). As an application, we come up with a criterion to determine whether a highly edge-connected graph with $\delta(G)\ge \delta_1+\cdots+ \delta_m$ (resp. $\Delta(G)\le \Delta_1+\cdots+ \Delta_m$) can be edge-decomposed into factors $G_1,\ldots, G_m$ satisfying $\delta(G_i)\ge \delta_i$ (resp. $\Delta(G_i)\le \Delta_i$) for all $i$ with $1\le i \le m$, provided that $\delta_1+\cdots+ \delta_m$ is divisible by an odd number $p$ and $\delta_i\ge p-1\ge 2$ (resp. $\Delta_1+\cdots+ \Delta_m$ is divisible by $p$ and $\Delta_i\ge p-1\ge 2$). For graphs of even order, we replace an odd-edge-connectivity condition. In particular, for the special case $m=2$, we refine the needed odd-edge-connectivity further by giving a sufficient odd-edge-connectivity condition for a graph $G$ to have a partial parity factor $F$ such that for each vertex $v$ with a given parity constraint, $| d_{F}(v)-\varepsilon d_G(v)|< 2$, and for all other vertices $v$, $| d_{F}(v)-\varepsilon d_G(v)|\le 1$, where $\varepsilon $ is a real number and $0< \varepsilon < 1$. Finally we introduce another application on the existence of almost even factorizations of odd-edge-connected graphs.