How large of a minimum degree does an $n$-vertex graph need before we are sure that it contains a perfect matching? Dirac's theorem states that a graph on an even number of vertices in which each vertex has degree at least $n/2$ has this property. In this short expository note, intended to be used in the classroom, we discuss how this statement generalizes to hypergraphs. In particular, we highlight an elegant connection between fractional perfect matchings in hypergraphs and a probabilistic inequality about nonnegative random variables, which was conjectured by Feige. We also present a very short self-contained proof of Feige's conjecture.
We prove that there is an absolute constant $c>0$ such that every connected vertex-transitive graph $G$ on $n \ge 3$ vertices contains a cycle of length at least $cn$. Moreover, every such graph with sufficiently large degree $d$ contains a cycle of length at least $(1-d^{-1/100})n$. This gives the first linear bound t...
Albertson's conjecture states that every graph of chromatic number r has crossing number at least that of the complete graph K_r. We prove the conjecture for all r. The proof combines new clique-sensitive edge bounds for critical graphs, maximum-clique sampling, and a reserved immersion construction. For graphs of orde...
Let be a graph with . Simonovits's theorem states that, if is edge‐critical, the unique largest ‐free subgraph of is its largest ‐partite subgraph, provided that is sufficiently large. We show that the same holds with replaced by the binomial random graph whenever is also strictly 2‐balanced and for some explicit con...
Ilay Hoshen, Wojciech Samotij· Proceedings of the London Ma...· 0 citations
We prove that, whenever $ p \ge n^{-1/2 + o(1)} $, with high probability $ G(n, p) $ admits a fractional triangle decomposition, that is, a non-negative weight function on its triangles for which the total weight of all triangles containing each edge is equal to 1. This bound on $ p $ is optimal up to the asymptotic er...
A graph is
perfect
if, for every induced subgraph, the chromatic number equals the size of its largest clique. In 1972, Lovász established a fundamental characterization of perfect graphs, showing that a graph is perfect if and only if, for every induced subgraph, the product of the size of the largest independen...
Let $G$ be a finite simple graph and let $S\subseteq V(G)$. We prove that the minimum number of vertices meeting every cycle that intersects $S$ is at most the maximum number of vertices of $S$ covered by a collection of vertex-disjoint cycles. This answers a question posed by Bowler, Ghorbani, Gut, Jacobs, and Reich [...
Han-Zhi Bai, Jin Yan· 0 citations
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