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A NOTE ON MONOTONICITY OF MIXED RAMSEY NUMBERS MARIA AXENOVICH AND JIHYEOK CHOI
 

Summary: A NOTE ON MONOTONICITY OF MIXED RAMSEY NUMBERS
MARIA AXENOVICH AND JIHYEOK CHOI
Abstract. For two graphs, G, and H, an edge-coloring of a complete graph is (G, H)-good if there
is no monochromatic subgraph isomorphic to G and no rainbow subgraph isomorphic to H in this
coloring. The set of number of colors used by some (G, H)-colorings of Kn is called a mixed-Ramsey
spectrum. This note addresses a fundamental question of whether the spectrum is an interval. It
is shown that the answer is "yes" if G is not a star and H does not contain a pendent edge.
1. Introduction
Let G and H be two graphs on fixed number of vertices. An edge coloring of a complete
graph, Kn, is called (G, H)-good if there is no monochromatic copy of G and no rainbow (totally
multicolored) copy of H in this coloring. This, sometimes called mixed-Ramsey coloring, is a
hybrid of classical Ramsey and anti-Ramsey colorings, [18, 6]. As shown by Jamison and West [15],
a (G, H)-good coloring of an arbitrarily large complete graph exists unless either G is a star or H
is a forest.
Let S(n; G, H) be the set of the number of colors, k, such that there is a (G, H)-good coloring
of Kn with k colors. We call S(n; G, H) a spectrum. Let max S(n; G, H), min S(n; G, H) be the
maximum, minimum number in S(n; G, H), respectively. The behavior of these functions was
studied in [2], [8], [1] and others. Note that if there is no restriction on a graph H, S(n; G, ) is
an interval [k, n
2 ], where k is the largest number such that rk-1(G) n, a classical multicolor

  

Source: Axenovich, Maria - Department of Mathematics, Iowa State University
Kovalev, Leonid - Department of Mathematics, Syracuse University

 

Collections: Mathematics