Combinatorial mathematics; proceedings of the second by D.A. Holton

By D.A. Holton

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Field Theory and Its Classical Problems (Carus Mathematical Monographs, Volume 19)

Submit yr observe: First released January 1st 1978
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Combinatorial mathematics; proceedings of the second Australian conference

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For completeness, just let me mention a related problem. A double blocking set is a collection B, with the property that every line intersects B in at least two points. Is it true that a double blocking set in PG(2, p), p prime, contains at least 3p points? Here the situation is even worse: It is only known (and a not too difficult exercise) for p = 2,3,4,5, 7. We now proceed to consider blocking sets of a special type. If B has size q + N then a line can contain at most N points of B. If such a line exists the blocking set is called of Redei type.

Proof Let G be a simple graph with average degree at least d. We first show that, as is well known, every such G contains a bipartite graph, whose minimum degree is at least d/4. To see this, observe that G has an induced subgraph with minimum degree at least d/2, since one can repeatedly delete vertices of degree smaller than d/2 from G, as long as there are such vertices; since this process increases the average degree, it must terminate in a nonempty subgraph G' with minimum degree at least d/2.

Alon, The linear arboricity of graphs, Israel J. Math. 62 (1988), 311-325. [4] N. Alon, Non-constructive proofs in Combinatorics, Proc. of the International Congress of Mathematicians, Kyoto 1990, Japan, Springer Verlag, Tokyo (1991), 1421-1429. [5] N. Alon, The strong chromatic number of a graph, Random Structures and Algorithms 3 (1992), 1-7. [6] N. Alon, Choice numbers of graphs; a probabilistic approach, Combina- torics, Probability and Computing 1 (1992), 107-114. [7] N. Alon, D. J. Kleitman, C.

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