The main conjecture on degree-\a0k+1k+1 vertices in minimal kk-extendable bipartite graphs

Let GG be a minimal kk-extendable bipartite graph, with k1k\geqslant 1, order nn, and size m=E(G)m=|E(G)|. Let Vk+1V_{k+1} be the set of vertices of degree precisely k+1k+1. Main conjecture.

Vk+122k1(mn+2k).|V_{k+1}|\geqslant \frac{2}{2k-1}(m-n+2k).

This conjecture is proposed as a strengthening of Lou's lower bound in terms of the order and size; its status is not resolved in the supplied text.

Sources & referencesView supporting material

Primary source

Amit Kumar Mallik, Ajit A. Diwan and Nishad Kothari, “Extremal minimal bipartite matching covered graphs”, arXiv:2404.06445 (2025).

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