Domination Density and an Imbalance Regime for Vizings Conjecture

Noah Hosking

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arXiv · 2509.10473

Domination Density and an Imbalance Regime for Vizings Conjecture

Noah Hosking

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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 \delta(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 \delta(G) \ne \delta(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.

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Noah Hosking. 2025-08-26. Domination Density and an Imbalance Regime for Vizings Conjecture. https://arxiv.org/abs/2509.10473

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