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Coloring Powers of Chordal Graphs
SIAM Journal on Discrete Mathematics, 2004Summary: We prove that the \(k\)th power \(G^{k}\) of a chordal graph \(G\) with maximum degree \(\Delta\) is \(O(\sqrt{k}\Delta^{(k+1)/2})\)-degenerate for even values of \(k\) and \(O(\Delta^{(k+1)/2})\)-degenerate for odd values. In particular, this bounds the chromatic number \(\chi(G^k)\) of the \(k\)th power of \(G\).
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1988
Let P be an undirected graph with vertices V and edges E. Fix an enumeration, {v1,v2,...,vn}, of V and let M(P) = {A ∈ Mn (ℂ)| = 0 if (vi,vj) ∉ E where ei is the standard orthonormal basis of ℂn. Mn (ℂ)+ is the set of positive semi-definite n × n matrices with complex entries.
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Let P be an undirected graph with vertices V and edges E. Fix an enumeration, {v1,v2,...,vn}, of V and let M(P) = {A ∈ Mn (ℂ)| = 0 if (vi,vj) ∉ E where ei is the standard orthonormal basis of ℂn. Mn (ℂ)+ is the set of positive semi-definite n × n matrices with complex entries.
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What Is between Chordal and Weakly Chordal Graphs?
2008An (h ,s ,t )-representation of a graph G consists of a collection of subtrees {S v | v *** V (G )} of a tree T , such that (i) the maximum degree of T is at most h , (ii) every subtree has maximum degree at most s , and (iii) there is an edge between two vertices in the graph if and only if the corresponding subtrees in T have at least t vertices in ...
Elad Cohen +3 more
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Distinct Classes of Complex Structural Variation Uncovered across Thousands of Cancer Genome Graphs
Cell, 2020Kevin Hadi +2 more
exaly