Abstract
The existence of a rainbow matching given a minimum color degree, proper coloring, or triangle-free host graph has been studied extensively. This paper generalizes these problems to edge colored graphs with given total color degree. In particular, we find that if a graph $G$ has total color degree $2mn$ and satisfies some other properties, then $G$ contains a matching of size $m$. These other properties include $G$ being triangle-free, $C_4$-free, properly colored, or large enough.
Highlights
Given a graph G, let V (G) denote the vertex set of G and E(G) denote the edge set of G
If X, Y ⊆ V (G), c(X, Y ) will denote the set of colors used on edges of the form xy, where x ∈ X, y ∈ Y
A rainbow matching is a matching where each edge receives a unique color within the matching
Summary
Given a graph G, let V (G) denote the vertex set of G and E(G) denote the edge set of G. This problem was generalized to find a function f such that any edge colored graph G with |V (G)| f (δ(G)) contains a rainbow matching of size δ(G). Local Anti-Ramsey theory is about the minimum k such that any coloring of Kn with δ(G) k contains a rainbow copy of H In this vein, Wang’s question can be posed as follows: given k, what is the smallest N such that any properly edge colored graph G with |V (G)| N and δ(G) k contains a rainbow matching of size k? If G is an edge colored graph on n vertices with d(G) 2mn, does G contain a rainbow matching of size m?.
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