Maximum flow and self-avoiding walk on bunkbed graphs
Pengfei Tang
TL;DR
This work analyzes two combinatorial processes on bunkbed graphs $G\times K_2$: maximum flow and self-avoiding walks. It extends the $p$-resistance framework to bunkbed graphs to derive a general inequality $\mathcal{R}_p(x_0,y_1)\ge\mathcal{R}_p(x_0,y_0)$ under reflection-symmetric capacities, and uses linear programming duality to translate this into a max-flow inequality $\mathbf{MF}(x_0,y_0) \ge \mathbf{MF}(x_0,y_1)$. For self-avoiding walks, it establishes a counterexample-driven landscape on ladders and proves a complete-graph result: for $G=K_n$, the count of SAWs from $(u,0)$ to $(v,1)$ eventually dominates those to $(u,0)$ from $(v,0)$, with explicit asymptotics showing the inequality holds for all $n\ge3$. The findings illuminate how structural features of $G$ (e.g., complete graphs vs. ladders) shape combinatorial flows and SAW counts on bunkbed graphs, highlighting both universal patterns and graph-dependent obstructions such as cut-edges.
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 $G$.