Ben Howerton | Searcharxiv

arXiv2026

A Two-Player Zero Forcing Game

We introduce a competitive two-player zero forcing game on a connected graph. Alice and Bob alternately seed white vertices or perform legal zero forces in their own colours, and each player seeks to minimise their own number of seeds. A force preservation rule prevents avoidable blocking of an opponent's established force. Because distinct continuations can be equally good for the player to move, optimal play is defined by a set-valued backward induction, and $Z_g(G)$ is the minimum total number of seeds among the resulting optimal outcomes. We prove that $Z_g(G)\geq Z(G)$, determine $Z_g$ for paths, cycles, stars, complete graphs, and complete bipartite graphs, and characterise the graphs with $Z_g(G)=2$ by an alternating two-chain forcing schedule. We also show that $Z_g$ is not minor-monotone and that edge subdivision can either increase or decrease the parameter. Exact computation verifies $Z_g(G)\leq2Z(G)$ through order nine.

math.CO↗

arXiv2026

Induced Subgraph Bounds on the Zero Forcing Number and Chromatic Consequences

Let $G$ be a graph with chromatic number $\chi(G)$, clique number $\omega(G)$ and zero forcing number $Z(G)$. We establish new lower bounds on $Z(G)$ in terms of induced triangle-free subgraphs. In particular, we show that if a graph $G$ contains an induced triangle-free subgraph $H$ with minimum degree $\delta(H) \ge 3$, then $Z(G)\ge\delta(H)+1$. As consequences, we prove that every triangle-free graph satisfies $\chi(G)\le\max\{3,Z(G)\}$ and obtain an application to planar graphs. Moreover, we prove that $\chi(G)\le \frac{Z(G)}{2}+2$ for every triangle-free graph.

math.CO↗