In a red-blue coloring of a nonempty graph, every edge is colored red or blue. If the resulting edge-colored graph contains a nonempty subgraph G without isolated vertices every edge of which is colored the same, then G is said to be monochromatic. For two nonempty graphs G and H without isolated vertices, the mono- chromatic Ramsey number mr(G,H) of G and H is the minimum integer n such that every red-blue coloring of Kn results in a monochromatic G or a monochromatic H. Thus, the standard Ramsey number of G and H is bounded below by mr(G,H). The monochromatic Ramsey numbers of graphs belonging to some common classes of graphs are studied. We also investigate another concept closely related to the standard Ram- sey numbers and monochromatic Ramsey numbers of graphs. For a fixed integer n ≥ 3, consider a nonempty subgraph G of order at most n con- taining no isolated vertices. Then G is a common monochromatic subgraph of Kn if every red-blue coloring of Kn results in a monochromatic copy of G. Furthermore, G is a maximal common monochromatic subgraph of Kn if G is a common monochromatic subgraph of Kn that is not a proper sub- graph of any common monochromatic subgraph of Kn. Let S(n) and S*(n) be the sets of common monochromatic subgraphs and maximal common monochromatic subgraphs of Kn, respectively. Thus, G ∈ S(n) if and only if R(G,G) = mr(G,G) ≤ n. We determine the sets S(n) and S*(n) for 3 ≤ n ≤ 8.
@article{bwmeta1.element.doi-10_7151_dmgt_1725,
author = {Eric Andrews and Futaba Fujie and Kyle Kolasinski and Chira Lumduanhom and Adam Yusko},
title = {On Monochromatic Subgraphs of Edge-Colored Complete Graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {34},
year = {2014},
pages = {5-22},
zbl = {1302.05115},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1725}
}
Eric Andrews; Futaba Fujie; Kyle Kolasinski; Chira Lumduanhom; Adam Yusko. On Monochromatic Subgraphs of Edge-Colored Complete Graphs. Discussiones Mathematicae Graph Theory, Tome 34 (2014) pp. 5-22. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1725/