William Staton

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A cubic triangle-free graph has a bipartite subgraph with at least 4/5 of the original edges. Examples show that this is a best possible result. It is easy to verify that the Petersen graph may be reduced to a bipartite graph by the removal of three of its 15 edges, and that this may not be done by the removal of two edges. Similarly, the Dodecahedron graph(More)
We investigate whether K,-free graphs with few repetitions in the degree sequence may have independence number o(n). We settle the cases r = 3 and r >/5, and give partial results for the very interesting case r=4 . In an earl ier art icle I-4] it is shown that t r iangle-free graphs in which no term of the degree sequence occurs more than twice must be(More)
Connected graphs with minimum degree δ and at least 2δ+ 1 vertices have paths with at least 2δ + 1 vertices. We provide a characterization of all such graphs which have no longer paths. Extremal problems involving paths and cycles have been considered since the infancy of graph theory. The question which interests us here is the question of what minimum(More)
Graphs with n + k vertices in which every set of n +j vertices induce a subgraph of maximum degree at least n are considered. For j = 1 and for k fairly small compared to n, we determine the minimum number of edges in such graphs. In investigating the size Ramsey number of a star KI,, versus a triangle K3, Erdiis [ 1,2] conjectured that for n 3 3 any graph(More)