Domination Density and an Imbalance Regime for Vizings Conjecture

Abstract

We develop a domination density framework for studying Vizings conjecture gamma(G square H) ge gamma(G)gamma(H). Recasting the conjecture in multiplicative density form we derive a bipartition imbalance sufficient condition for certain graph pairs. For bipartite G we introduce a constructive imbalance amplification argument: if a minimum dominating set of G is sufficiently concentrated on one side of the bipartition relative to δ(H) then gamma(G square H) + tau_X |V(H)| ge gamma(G)gamma(H) where tau_X depends explicitly on the local domination concentration. In particular whenever delta(G) > delta(H)or for when G and H are bipartite with δ(G) \ne δ(H) such a concentration must occur. This yields an explicit additive deficit bound within the same domination density regime. We further show that domination-reducing leaf deletions preserve Vizings inequality. Consequently for bipartite graphs satisfying delta(G) > delta(H) the conjecture reduces to understanding the stability of Vizings inequality under domination-neutral leaf deletions.