Vizings Conjecture: A Density-Based Re-framing Applied to Bipartite Graphs
Noah Hosking
TL;DR
This work reframes Vizing's conjecture on the domination number of Cartesian products as a density inequality $ρ_{G□H} \ge ρ_G ρ_H$, enabling analytic bounds to certify the conjecture in large bipartite regimes. It proves the conjecture for bipartite graphs with sufficiently uneven partitions and introduces a constructive bound $γ(G□H) + m_X^{\ast}|V(H)| \ge γ(G) γ(H)$ via structural transformations. Extending to balanced $k$-regular bipartite graphs, the authors propose a conjectural bound $γ(G) \le 2\left\lceil \frac{n}{k} \right\rceil$ that, if valid for $k \ge 7$, yields Vizing's inequality for all such graphs except a finite set, and they analyze a dimension obstruction through the biadjacency matrix to explain one-sided coverings. Collectively, the density framework unifies combinatorial and algebraic perspectives, offering a path to certify vast families and identify a finite set of remaining boundary cases, with future work aimed at sharpening bounds and computing the finite remainder. The results have potential implications for understanding domination in Cartesian products and guiding algorithmic verification of graph product conjectures.
Abstract
We reformulate Vizing's conjecture γ(G\square H) \ge γ(G)γ(H) in terms of normalised domination density and use analytic bounds to delineate regimes where it holds. The conjecture is verified for all bipartite pairs with sufficiently uneven bipartitions. We establish γ(G \square H) + m_X^{\ast}|V(H)| \ge γ(G)γ(H) as a new constructive inequality, extending validity under controlled structural transformations for certain bipartite graphs. Finally, assuming a conjectural k-regular domination number bound, the conjecture holds for all balanced k-regular bipartite graphs with k\ge7, leaving only finitely many small cases unresolved.