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Preprint Oct 2026

A Counterexample to the Alon-Krivelevich Conjecture and a New Lower Bound for Choosability of Bipartite Graphs

For an integer $k\ge 2$ and $r^2>2(2k-1)\ln (2k-1)$, we construct a bipartite graph $G$ whose two vertex classes are disjoint copies of the family of all $k$-subsets of a $(2k-1)$-element set, two vertices being adjacent when their sets intersect in at most $r$ elements. Assigning each vertex its corresponding subset a...

Jun-Ran Yu, G. Yu · 0 citations
Preprint Oct 2026

Separating Path Systems of Size at most $7.75n$

A family of paths in a graph $G$ strongly separates the edges of $G$ if for every ordered pair of distinct edges $(e,f)$ some path in the family contains $e$ and avoids $f$; the minimum size of such a family is denoted by $\operatorname{ssp}(G)$. Bonamy, Botler, Dross, Naia, and Skokan proved, for every $n$-vertex grap...

Daniel W. Cranston, Jared Noyman, G. Yu · 0 citations

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