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AVOIDING RAINBOW INDUCED SUBGRAPHS IN EDGE-COLORINGS CHELSEA SACKETT AND MARIA AXENOVICH
 

Summary: AVOIDING RAINBOW INDUCED SUBGRAPHS IN EDGE-COLORINGS
CHELSEA SACKETT AND MARIA AXENOVICH
Abstract. Let H be a fixed graph on k vertices. For an edge-coloring c of H, we say that H is
rainbow, or totally multicolored if c assigns distinct colors to all edges of H. We show, that it is easy
to avoid rainbow induced graphs H. Specifically, we prove that for any graph H (with some notable
exceptions), and for any graphs G, G = H, there is an edge-coloring of G with k colors which contains
no induced rainbow subgraph isomorphic to H. This demonstrates that, in a sense, induced subgraphs
do not have "anti-Ramsey"-type properties.
1. Introduction
1
Let G = (V, E) be a graph. Let c : E(G) [k] be an edge-coloring of G. We say that G
is monochromatic under c if all edges have the same color, and we say that G is rainbow or totally
multicolored if all edges of G have distinct colors. A graph G = (V , E) is an induced subgraph of
G if V V and e E if and only if e E. Ramsey, see [12], has shown that the monochromatic
subgraphs are unavoidable in colorings of large complete graphs with fixed numbers of colors. Erdos,
Simonovits and Sos, see [7], proved that the totally multicolored subgraphs are unavoidable if the
number of colors used on the graphs is large enough. When considered colored subgraphs are induced,
the situation becomes more subtle. Deuber, see [3] proved the induced variant of Ramsey theorem by
showing that for any two graphs H1 and H2, there is a graph G such that in any coloring of edges of
G in two colors 1 and 2, there is an induced subgraph of G isomorphic to Hi which is monochromatic

  

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

 

Collections: Mathematics