On the chromatic number of regular graphs of matrix algebras
On the chromatic number of regular graphs of matrix algebras
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DOI：
10.1016/j.laa.2015.02.030
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发表时间：
2015-06
影响因子：
1.1
通讯作者：
István Tomon
中科院分区：
文献类型：
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作者：
István Tomon
设R是一个环，Z (R)是R的零因子的集合，R的正则图，记为Γ (R)是顶点集R∈Z (R)的图，当X+ Y∈Z (R)时{X， Y}是一条边。证明了Γ （mn (F q)）的色数至少是（q/4）⌊n/2⌋，其中mn （F q）是F q上n× n个矩阵组成的环，其中q是奇素数幂。这证明了Γ （mn (F p alg)）的色数是无限的，回答了BCC22中提出的一个问题。
Let R be a ring and Z (R) be the set of zero divisors of R. The regular graph of R, denoted by Γ (R) is the graph with vertex set R∖ Z (R) and {X, Y} is an edge if X+ Y∈ Z (R). We prove that the chromatic number of Γ (M n (F q)) is at least (q/4)⌊ n/2⌋, where M n (F q) is the ring of n× n matrices over F q, q being an odd prime power. This proves that the chromatic number of Γ (M n (F p alg)) is infinite, answering a case of a question posed in BCC22.