A graph coloring algorithm that immediately colors the vertices taken from a list without looking ahead or changing colors already assigned is called "on-line coloring." The properties of on-line colorings are investigated in several classes of graphs. In many cases w e find on-line colorings that use no more colors than some function of the largest clique size of the graph. We show that the first fit on-line coloring has an absolute performance ratio of two for the complement of chordal graphs. We prove an upper bound for the performance ratio of the first fit coloring on interval graphs. It is also shown that there are simple families resisting any on-line algorithm: no on-line algorithm can color all trees by a bounded number of colors.

An adjacent vertex distinguishing edge-coloring of a simple graph G is a proper edge-coloring of G such that no pair of adjacent vertices meets the same set of colors. The minimum number of colors χ a (G) required to give G an adjacent vertex distinguishing coloring is studied for graphs with no isolated edge. We prove χ a (G) ≤ 5 for such graphs with maximum degree Δ(G) = 3 and prove χ a (G) ≤ Δ(G) + 2 for bipartite graphs. These bounds are tight. For k-chromatic graphs G without isolated edges we prove a weaker result of the form χ a (G) = Δ(G) + O(log k).

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We conjecture that for any fixed r and sufficiently large n, there is a monochromatic Hamiltonian Bergecycle in every (r − 1)-coloring of the edges of K (r) n , the complete r-uniform hypergraph on n vertices. We prove the conjecture for r = 3, n 5 and its asymptotic version for r = 4. For general r we prove weaker forms of the conjecture: there is a Hamiltonian Berge-cycle in (r − 1)/2 -colorings of K (r) n for large n; and a Berge-cycle of order (1 − o(1))n in (r − log 2 r )-colorings of K (r) n . The asymptotic results are obtained with the Regularity Lemma via the existence of monochromatic connected almost perfect matchings in the multicolored shadow graph induced by the coloring of K (r) n .

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It is easily shown that every digraph with m edges has a directed cut of size at least m/4, and that 1/4 cannot be replaced by any larger constant. We investigate the size of a largest directed cut in acyclic digraphs, and prove a number of related results concerning cuts in digraphs and acyclic digraphs.

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