54 research outputs found
Choice Number and Energy of Graphs
The energy of a graph G, denoted by E(G), is defined as the sum of the
absolute values of all eigenvalues of G. It is proved that E(G)>=
2(n-\chi(\bar{G}))>= 2(ch(G)-1) for every graph G of order n, and that E(G)>=
2ch(G) for all graphs G except for those in a few specified families, where
\bar{G}, \chi(G), and ch(G) are the complement, the chromatic number, and the
choice number of G, respectively.Comment: to appear in Linear Algebra and its Application
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
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