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An on-line vertex coloring algorithm receives the vertices of a graph in some externally determined order, and, whenever a new vertex is presented, the algorithm also learns to which of the previously presented vertices the new vertex is adjacent. As each vertex is received, the algorithm must make an irrevocable choice of a color to assign the new vertex,… (More)

For a graph H, let Forb(H) be the class of graphs that do not induce H, and let P5 be the path on five vertices. In this article, we answer two questions of Gyrfs and Lehel. First, we show that there exists a function f(w) such that for any graph G E Forb(Ph), the on-line coloring algorithm First-Fit uses at most f(w(G)) colors on G, where w(G) is the… (More)

All \labelings" discussed in this paper are vertex labelings. We call a labeling l of G an N-labeling if there exists a connected induced subgraph H of G with P x2V (H) l(x) = i for every integer i, 1 i N. Then c (G) is the largest integer N such that G has an N-labeling. A labeling l of G is called irredundant if, for all connected induced subgraphs H 1… (More)

A graph G is balanced if the maximum ratio of edges to vertices, taken over all subgraphs of G, occurs at G itself. This note uses the max-ow/min-cut theorem to prove a good characterization of balanced graphs. This characterization is then applied to some results on how balanced graphs may be combined to form a larger balanced graph. In particular, we show… (More)

Continuing work by BB acso and Tuza and Cozzens and Kelleher, we investigate dominating sets which induce subgraphs with small clique covering number, or small independence number. We show that if a graph G is connected and contains no induced subgraph isomorphic to P 6 or H t (the graph obtained by subdividing each edge of K 1;t ; t 3), then G has a… (More)