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991.
Eldar Fischer 《Journal of Graph Theory》1999,31(4):275-282
The Hajnal-Szemerédi Theorem [Hajnal & Szemerédi, Colloq Math Soc J Bolyai, 1970] states that a graph with hk vertices and minimum degree at least (h − 1)k contains k vertex disjoint copies of Kh. Its proof is not algorithmic. Here, we present an algorithm that, for a fixed h, finds in such a graph k − C(h) vertex disjoint copies of Kh in polynomial time in k, C(h) being a constant depending on h only. The proof suggests a variant of the theorem for h-partite graphs, which is conjectured here and proven in a slightly weaker form in some special cases. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 275–282, 1999 相似文献
992.
Bruce Reed 《Journal of Graph Theory》1999,31(2):149-153
We present a conjecture concerning list colourings and prove a weakened form of it. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 149–153, 1999 相似文献
993.
In this article, we introduce the new notion of acyclic improper colorings of graphs. An improper coloring of a graph is a vertex-coloring in which adjacent vertices are allowed to have the same color, but each color class Vi satisfies some condition depending on i. Such a coloring is acyclic if there are no alternating 2-colored cycles. We prove that every outerplanar graph can be acyclically 2-colored in such a way that each monochromatic subgraph has degree at most five and that this result is best possible. For planar graphs, we prove some negative results and state some open problems. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 97–107, 1999 相似文献
994.
995.
Given a BIBD S = (V, B), its 1-block-intersection graph Gs has as vertices the elements of B; two vertices B1, B2 ∈ B are adjacent in Gs if |B1 ∩ B2| = 1. If S is a triple system of arbitrary index λ, it is shown that GS is hamiltonian. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 243-246, 1999 相似文献
996.
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998.
The concept of a perfect coloring, introduced by P. Delsarte, generalizes the concept of completely regular code. We study the perfect 3-colorings (also known as the equitable partitions into three parts) on 6-regular graphs of order 9. A perfect n-colorings of a graph is a partition of its vertex set. It splits vertices into n parts A1, A2,...,An such that for all i,j∈{1,2,...,n}, each vertex of Ai is adjacent to aij vertices of Aj. The matrix A =(aij)n×n is called quotient matrix or parameter matrix. In this article, we start by giving an algorithm to find all different types of 6-regular graphs of order 9. Then, we classify all the realizable parameter matrices of perfect 3-colorings on 6-regular graphs of order 9. 相似文献
999.
Xiaofeng XUE 《Frontiers of Mathematics in China》2018,13(3):667-690
We are concerned with the susceptible-infective-removed (SIR) model with random transition rates on complete graphs C n with n vertices. We assign independent and identically distributed (i.i.d.) copies of a positive random variable ξ on each vertex as the recovery rates and i.i.d. copies of a positive random variable ρ on each edge as the edge infection weights. We assume that a susceptible vertex is infected by an infective one at rate proportional to the edge weight on the edge connecting these two vertices while an infective vertex becomes removed with rate equals the recovery rate on it, then we show that the model performs the following phase transition when at t = 0 one vertex is infective and others are susceptible. There exists λ c > 0 such that when λ < λ c ; the proportion r∞ of vertices which have ever been infective converges to 0 weakly as n → +∞ while when λ > λ c ; there exist c(λ) > 0 and b(λ) > 0 such that for each n ≥ 1 with probability p ≥ b(λ); the proportion r∞ ≥ c(λ): Furthermore, we prove that λ c is the inverse of the production of the mean of ρ and the mean of the inverse of ξ. 相似文献
1000.