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Title: Generalizations and Strengthenings of Ryser's Conjecture
Ryser's conjecture says that for every $$r$$-partite hypergraph $$H$$ with matching number $$\nu(H)$$, the vertex cover number is at most $$(r-1)\nu(H)$$.  This far-reaching generalization of König's theorem is only known to be true for $$r\leq 3$$, or when $$\nu(H)=1$$ and $$r\leq 5$$.  An equivalent formulation of Ryser's conjecture is that in every $$r$$-edge coloring of a graph $$G$$ with independence number $$\alpha(G)$$, there exists at most $$(r-1)\alpha(G)$$ monochromatic connected subgraphs which cover the vertex set of $$G$$.   We make the case that this latter formulation of Ryser's conjecture naturally leads to a variety of stronger conjectures and generalizations to hypergraphs and multipartite graphs.  Regarding these generalizations and strengthenings, we survey the known results, improving upon some, and we introduce a collection of new problems and results.  more » « less
Award ID(s):
1954170
PAR ID:
10335796
Author(s) / Creator(s):
; ; ;
Date Published:
Journal Name:
The Electronic Journal of Combinatorics
Volume:
28
Issue:
4
ISSN:
1077-8926
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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