On maximum $k$-edge-colorable subgraphs of bipartite graphs

TL;DR

This paper investigates bounds on maximum $k$-edge-colorable subgraphs in bipartite graphs, extending known results from cubic graphs and establishing new inequalities relating these subgraphs for bipartite graphs.

Contribution

It proves a new inequality relating $ u_k(G)$, $ u_{k-i}(G)$, and $ u_{k+i}(G)$ specifically for bipartite graphs, generalizing previous results.

Findings

Established that $ u_{k}(G) igg floor rac{ u_{k-i}(G) + u_{k+i}(G)}{2}$ for bipartite graphs.

Extended known bounds from cubic graphs to bipartite graphs.

Provided a new inequality linking maximum $k$-edge-colorable subgraphs across different $k$ values.

Abstract

If $k\geq 0$, then a $k$-edge-coloring of a graph $G$ is an assignment of colors to edges of $G$ from the set of $k$ colors, so that adjacent edges receive different colors. A $k$-edge-colorable subgraph of $G$ is maximum if it is the largest among all $k$-edge-colorable subgraphs of $G$. For a graph $G$ and $k\geq 0$, let $\nu_{k}(G)$ be the number of edges of a maximum $k$-edge-colorable subgraph of $G$. In 2010 Mkrtchyan et al. proved that if $G$ is a cubic graph, then $\nu_2(G)\leq \frac{|V|+2\nu_3(G)}{4}$. This result implies that if the cubic graph $G$ contains a perfect matching, in particular when it is bridgeless, then $\nu_2(G)\leq \frac{\nu_1(G)+\nu_3(G)}{2}$. One may wonder whether there are other interesting graph-classes, where a relation between $\nu_2(G)$ and $\frac{\nu_1(G)+\nu_3(G)}{2}$ can be proved. Related with this question, in this paper we show that $\nu_{k}(G) \geq \frac{\nu_{k-i}(G) + \nu_{k+i}(G)}{2}$ for any bipartite graph $G$, $k\geq 0$ and $i=0,1,...,k$.