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101.
Two n‐vertex hypergraphs G and H pack, if there is a bijection such that for every edge , the set is not an edge in H. Extending a theorem by Bollobás and Eldridge on graph packing to hypergraphs, we show that if and n‐vertex hypergraphs G and H with with no edges of size 0, 1, and n do not pack, then either
- one of G and H contains a spanning graph‐star, and each vertex of the other is contained in a graph edge, or
- one of G and H has edges of size not containing a given vertex, and for every vertex x of the other hypergraph some edge of size does not contain x.
102.
A graph G is equitably k‐choosable if for every k‐list assignment L there exists an L‐coloring of G such that every color class has at most vertices. We prove results toward the conjecture that every graph with maximum degree at most r is equitably ‐choosable. In particular, we confirm the conjecture for and show that every graph with maximum degree at most r and at least r3 vertices is equitably ‐choosable. Our proofs yield polynomial algorithms for corresponding equitable list colorings. 相似文献
103.
The order dimension of suborders of the Boolean lattice
is considered. It is shown that the suborder of
consisting of levels s and s+1 has dimension O(\log n/log log n). This improves a bound in [1]. 相似文献
104.
O. V. Borodin D. G. Fon‐Der Flaass A. V. Kostochka A. Raspaud E. Sopena 《Journal of Graph Theory》2002,40(2):83-90
The acyclic list chromatic number of every planar graph is proved to be at most 7. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 83–90, 2002 相似文献
105.
A. V. Kostochka 《Combinatorica》1982,2(2):187-192
Letf(n) denote the minimal number of edges of a 3-uniform hypergraphG=(V, E) onn vertices such that for every quadrupleY ⊂V there existsY ⊃e ∈E. Turán conjectured thatf(3k)=k(k−1)(2k−1). We prove that if Turán’s conjecture is correct then there exist at least 2
k−2 non-isomorphic extremal hypergraphs on 3k vertices. 相似文献
106.
A graph G is k-critical if it has chromatic number k, but every proper subgraph of G is (k?1)-colorable. Let f k (n) denote the minimum number of edges in an n-vertex k-critical graph. In a very recent paper, we gave a lower bound, f k (n)≥(k, n), that is sharp for every n≡1 (mod k?1). It is also sharp for k=4 and every n≥6. In this note, we present a simple proof of the bound for k=4. It implies the case k=4 of two conjectures: Gallai in 1963 conjectured that if n≡1 (mod k?1) then \(f_k (n)\tfrac{{(k + 1)(k - 2)n - k(k - 3)}} {{2(k - 1)}}\) , and Ore in 1967 conjectured that for every k≥4 and \(n \geqslant k + 2,f_k (n + k - 1) = f(n) + \tfrac{{k - 1}} {2}(k - \tfrac{2} {{k - 1}})\) . We also show that our result implies a simple short proof of Grötzsch’s Theorem that every triangle-free planar graph is 3-colorable. 相似文献