181 research outputs found
Large rainbow matchings in large graphs
A \textit{rainbow subgraph} of an edge-colored graph is a subgraph whose
edges have distinct colors. The \textit{color degree} of a vertex is the
number of different colors on edges incident to . We show that if is
large enough (namely, ), then each -vertex graph with
minimum color degree at least contains a rainbow matching of size at least
Harmonious Coloring of Trees with Large Maximum Degree
A harmonious coloring of is a proper vertex coloring of such that
every pair of colors appears on at most one pair of adjacent vertices. The
harmonious chromatic number of , , is the minimum number of colors
needed for a harmonious coloring of . We show that if is a forest of
order with maximum degree , then h(T)=
\Delta(T)+2, & if $T$ has non-adjacent vertices of degree $\Delta(T)$;
\Delta(T)+1, & otherwise.
Moreover, the proof yields a polynomial-time algorithm for an optimal
harmonious coloring of such a forest.Comment: 8 pages, 1 figur
- …