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• # Artykuł - szczegóły

## Discussiones Mathematicae Graph Theory

2014 | 34 | 1 | 5-22

## On Monochromatic Subgraphs of Edge-Colored Complete Graphs

EN

### Abstrakty

EN
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.

EN

5-22

wydano
2014-02-01
online
2014-02-14

### Twórcy

autor
• Department of Mathematics Western Michigan University Kalamazoo, MI 49008 USA
autor
• Graduate School of Mathematics Nagoya University Nagoya, Japan 464-8602
autor
• Department of Mathematics Western Michigan University Kalamazoo, MI 49008 USA
autor
• Department of Mathematics Western Michigan University Kalamazoo, MI 49008 USA
autor
• Department of Mathematics Western Michigan University Kalamazoo, MI 49008 USA

### Bibliografia

• [1] G. Chartrand, L. Lesniak and P. Zhang, Graphs and Digraphs (Chapman and Hall/CRC, Boca Raton, FL., 2010).
• [2] S.P. Radziszowski, Small Ramsey numbers, Electron. J. Combin. (2011) DS1.