Spanning caterpillar in biconvex bipartite graphs

TL;DR

The paper addresses whether every connected biconvex bipartite graph $G=(A,B,E)$ contains a spanning caterpillar. It provides a constructive proof based on a biconvex $S$-ordering and case analysis, introducing a residual path $P$ and a vertex-replacement technique to ensure all vertices lie on or attach to $P$. The main contribution is the existential result of a spanning caterpillar, with the corollary that the burning number satisfies $b(G) \le \lceil \sqrt{n} \rceil$ for this class, confirming the burning number conjecture for biconvex bipartite graphs and for subclasses such as bipartite permutation graphs and chain graphs. The result offers a structural tool for further graph-theoretic investigations of biconvex bipartite graphs and highlights contrasts with general convex bipartite graphs that may lack spanning caterpillars.

Abstract

A bipartite graph $G=(A, B, E)$ is said to be a biconvex bipartite graph if there exist orderings $<_A$ in $A$ and $<_B$ in $B$ such that the neighbors of every vertex in $A$ are consecutive with respect to $<_B$ and the neighbors of every vertex in $B$ are consecutive with respect to $<_A$. A caterpillar is a tree that will result in a path upon deletion of all the leaves. In this note, we prove that there exists a spanning caterpillar in any connected biconvex bipartite graph. Besides being interesting on its own, this structural result has other consequences. For instance, this directly resolves the burning number conjecture for biconvex bipartite graphs.

Figures (1)

• Figure 1: A convex bipartite graph having no spanning caterpillar

Theorems & Definitions (10)

• Lemma 1.1: Driscoll2017TreesSixCordial
• Theorem 1.3: Abbas2000Biconvex
• Theorem 1.4: Abbas2000Biconvex
• Theorem 1.5
• proof
• Claim 1
• proof
• Conjecture 3.1: Bonato2016Burning
• Theorem 3.2: Hiller2021BurningCaterpillar
• Corollary 3.3