Maximum flow and self-avoiding walk on bunkbed graphs

TL;DR

This paper explores maximum flow and self-avoiding walk models on bunkbed graphs, establishing flow inequalities under symmetric capacities and analyzing walk counts, with results for complete graphs and open questions for general cases.

Contribution

It proves flow inequalities for bunkbed graphs with symmetric capacities and demonstrates self-avoiding walk inequalities for large complete graphs, raising new questions for broader classes.

Findings

Maximum flow from (u,0) to (v,0) is at least as large as to (v,1) under symmetric capacities.

For large complete graphs, there are more self-avoiding walks from (u,0) to (v,1) than to (v,0).

Counterexamples exist, and open questions remain for general graphs and non-cut-edge pairs.

Abstract

We consider two combinatorial models on bunkbed graphs: maximum flow and self-avoiding walks. A bunkbed graph is defined as the Cartesian product $G\times K_2$, where $G$ is a finite graph and $K_2$ is the complete graph on two vertices, labelled $0$ and $1$. For the maximum flow problem, we show that if the bunkbed graph $G\times K_2$ has non-negative, reflection-symmetric edge capacities, then for any $u, v\in V(G)$, the maximum flow strength from $(u,0)$ to $(v,0)$ in $G\times K_2$ is at least as large as that from $(u,0)$ to $(v,1)$. For the self-avoiding walk model on a bunkbed graph $G\times K_2$, we investigate whether there are more self-avoiding walks from $(u,0)$ to $(v,1)$ than from $(u,0)$ to $(v,0)$. We prove that this holds when $G=K_n$ is a complete graph and $n$ is sufficiently large. Additionally, we provide examples where the statement does not holds and pose the question of whether it remains true when $\{u,v\}$ is not a cut-edge of $G$.