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The complete closure of a graph
Journal of Graph Theory, 1993AbstractWe define the complete closure number cc(G) of a graph G of order n as the greatest integer k ≤ 2n − 3 such that the kth Bondy‐Chvátal closure Clk(G) is complete, and give some necessary or sufficient conditions for a graph to have cc(G) = k.
Ralph J. Faudree +3 more
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2013
Let T 1, T 2,…, T k be spanning trees in a graph G. If for any two vertices x, y of G, the paths from x to y in T 1, T 2,…, T k are vertex-disjoint except end vertices x and y, then T 1, T 2,…, T k are called completely independent spanning trees in G. In 2001, Hasunuma gave a conjecture that there are k completely independent spanning trees in any 2k ...
Kung-Jui Pai +3 more
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Let T 1, T 2,…, T k be spanning trees in a graph G. If for any two vertices x, y of G, the paths from x to y in T 1, T 2,…, T k are vertex-disjoint except end vertices x and y, then T 1, T 2,…, T k are called completely independent spanning trees in G. In 2001, Hasunuma gave a conjecture that there are k completely independent spanning trees in any 2k ...
Kung-Jui Pai +3 more
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Combinatorica, 2007
Complete partitions of a graph are vertex partitions such that any two classes are related by an arc. The authors compute tight lower and upper bounds for the maximum number of classes in a complete partition. A technique used is that of finding the largest integer \(\beta(G)\) such that there exists a subgraph \(H\subseteq G\) with maximum degree at ...
Magnús M. Halldórsson +3 more
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Complete partitions of a graph are vertex partitions such that any two classes are related by an arc. The authors compute tight lower and upper bounds for the maximum number of classes in a complete partition. A technique used is that of finding the largest integer \(\beta(G)\) such that there exists a subgraph \(H\subseteq G\) with maximum degree at ...
Magnús M. Halldórsson +3 more
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Complete multipartite decompositions of complete graphs and complete n-partite graphs
Applied Mathematics-A Journal of Chinese Universities, 2003zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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The Coarseness of the Complete Graph
Canadian Journal of Mathematics, 1968The coarseness, c(G), of a graph G is the maximum number of edge-disjoint, non-planar subgraphs of G. We consider only the complete graph, Kp, on p vertices here. For p = 3r, Erdös conjectured that the coarseness was , but it has been shown (1) that1where square brackets denote integer part.
Guy, R. K., Beineke, L. W.
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Decompositions of Complete Graphs
Bulletin of the London Mathematical Society, 2000Summary: If \(s_1,s_2,\dots, s_t\) are integers such that \(n-1= s_1+ s_2+\cdots+ s_t\) and such that for each \(i\) \((1\leq i\leq t)\), \(2\leq s_i\leq n-1\) and \(s_in\) is even, then \(K_n\) can be expressed as the union \(G_1\cup G_2\cup\cdots\cup G_t\) of \(t\) edge-disjoint factors, where for each \(i\), \(G_i\) is \(s_i\)-connected.
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Canadian Journal of Mathematics, 1967
In our paper “Magic graphs” (1) we showed that every complete graph Kn with n ⩾ 5 is “magic,” i.e., if the vertex set is indicated {vi} and if eij is the edge joining vi and vj, i ≠ j , then there exists a function α(eij) such that the set {α(eij)} consists of distinct positive rational integers and the vertex sums1have a constant value σ(α) for k = 1,
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In our paper “Magic graphs” (1) we showed that every complete graph Kn with n ⩾ 5 is “magic,” i.e., if the vertex set is indicated {vi} and if eij is the edge joining vi and vj, i ≠ j , then there exists a function α(eij) such that the set {α(eij)} consists of distinct positive rational integers and the vertex sums1have a constant value σ(α) for k = 1,
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Multidecompositions of line graphs of complete graphs
Discrete Mathematics, Algorithms and Applications, 2019By a [Formula: see text]-decomposition of a graph [Formula: see text] we mean a decomposition of [Formula: see text] into [Formula: see text] copies of [Formula: see text] [Formula: see text] copies of [Formula: see text] and [Formula: see text] copies of [Formula: see text], where [Formula: see text] are non-negative integers.
S. Ganesamurthy 0001 +2 more
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Graphs omitting sums of complete graphs
Journal of Graph Theory, 1997There is a finite number of countable graphs omitting \(G\) such that every such graph is embedded into one of them if \(G\) is the vertex disjoint union of complete graphs. This was conjectured by Pach and the reviewer. The required number is determined in some cases.
Gregory L. Cherlin, Niandong Shi
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On the vulnerability of permutation graphs of complete and complete bipartite graphs
1991The integrity of a graph \(G\) is defined as \(\min\{| S|+m(G-S)\}\) taken over all subsets \(S\) of \(V(G)\), where \(m(G-S)\) is the order of the largest component of \(G-S\). The toughness of \(G\) is defined as \(\min\{| S|/w(G-S)\}\) taken over all disconnecting subsets \(S\) of \(G\), where \(w(G-S)\) is the number of components of \(G-S\).
Guichard, D. +2 more
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