51 research outputs found
On the number of symbols that forces a transversal
Akbari and Alipour conjectured that any Latin array of order with at
least symbols contains a transversal. We confirm this conjecture for
large , and moreover, we show that symbols suffice.Comment: 6 pages, 1 figur
A conjecture on critical graphs and connections to the persistence of associated primes
We introduce a conjecture about constructing critically (s+1)-chromatic
graphs from critically s-chromatic graphs. We then show how this conjecture
implies that any unmixed height two square-free monomial ideal I, i.e., the
cover ideal of a finite simple graph, has the persistence property, that is,
Ass(R/I^s) \subseteq Ass(R/I^{s+1}) for all s >= 1. To support our conjecture,
we prove that the statement is true if we also assume that \chi_f(G), the
fractional chromatic number of the graph G, satisfies \chi(G) -1 < \chi_f(G) <=
\chi(G). We give an algebraic proof of this result.Comment: 11 pages; Minor changes throughout the paper; to appear in Discrete
Math
Vertex covers by monochromatic pieces - A survey of results and problems
This survey is devoted to problems and results concerning covering the
vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles
and other objects. It is an expanded version of the talk with the same title at
the Seventh Cracow Conference on Graph Theory, held in Rytro in September
14-19, 2014.Comment: Discrete Mathematics, 201
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