MINIMUM DEGREES OF MINIMAL RAMSEY GRAPHS
MINIMUM DEGREES OF MINIMAL RAMSEY GRAPHS
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最小拉姆齐图的最小度
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Raj Raina
中科院分区：
文献类型：
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作者：
Raj Raina
对于图F和H，我们说如果F的每个2颜色包含H的单色副本，则F对于H。 F'是H. Burr，ErdőS和Lovász的Ramsey，定义为S（H）是所有Ramsey H-Minimal图的最小f，d定义了HT，d是T+ 1个顶点。图上图顶点和一个d度的顶点D。令人惊讶的是，S（HT，2）= 4都小得多。 h是h（h）=2δ（h）-1的最小h Ramsey Simple。如果没有孤立的顶点，我们很简单。简单的。
For graphs F and H, we say F is Ramsey for H if every 2-coloring of the edges of F contains a monochromatic copy of H. The graph F is Ramsey H-minimal if there is no proper subgraph F ′ of F so that F ′ is Ramsey for H. Burr, Erdős, and Lovász defined s(H) to be the minimum degree of F over all Ramsey H-minimal graphs F . Define Ht,d to be a graph on t+ 1 vertices consisting of a complete graph on t vertices and one additional vertex of degree d. We show that s(Ht,d) = d 2 for all values 1 < d ≤ t; it was previously known that s(Ht,1) = t− 1, so it is surprising that s(Ht,2) = 4 is much smaller. We also make some further progress on some sparser graphs. Fox and Lin observed that s(H) ≥ 2δ(H)−1 for all graphs H, where δ(H) is the minimum degree of H; a graph H with s(H) = 2δ(H)−1 is called Ramsey simple. Szabó, Zumstein, and Zürcher were the first to ask which graphs are Ramsey simple, and conjectured that all bipartite graphs without isolated vertices are. Fox, Grinshpun, Liebenau, Person, and Szabó further conjectured that all triangle-free graphs without isolated vertices are Ramsey simple. We show that d-regular 3-connected triangle-free graphs, with one extra technical constraint, are Ramsey simple.