On the null structure of bipartite graphs without cycles of length a multiple of 4

TL;DR

This paper investigates the null space structure of bipartite graphs without cycles of length multiple of 4, revealing decompositions, formulas, and rank relations that extend known results for trees.

Contribution

It introduces a novel decomposition of such graphs into subgraphs with specific properties and generalizes rank and independence results beyond trees.

Findings

Decomposition into $C_N(G)$ and $C_S(G)$ with distinct properties

Rank of the graph equals twice its matching number

Maximum independent sets intersect at the null space support

Abstract

In this work we study the null space of bipartite graphs without cycles of length multiple of $4$, and its relation to structural properties. We decompose them into two subgraphs: $C_N(G)$ and $C_S(G)$. $C_N(G)$ has perfect matching and its adjacency matrix is nonsingular. $C_S(G)$ has a unique maximum independent set and the dimension of its null space equals the dimension of the null space of $G$. Even more, we show that the fundamental spaces of $G$ are the direct sum of the fundamental spaces of $C_N(G)$ and $C_S(G)$. We also obtain formulas relating the independence number and the matching number of a $C_{4k}$-free bipartite graph with $C_N(G)$ and $C_S(G)$, and the dimensions of the fundamental spaces. Among other results, we show that the rank of a $C_{4k}$-free bipartite graph is twice its matching number, generalizing a result for trees due to Bevis et al \cite{bevis1995ranks}, and Cvetkovi\'c and Gutman \cite{D1972}. About maximum independent sets, we show that the intersection of all maximum independent sets of a $C_{4k}$-free bipartite graph coincides with the support of its null space.