Displaying similar documents to “Euler's idoneal numbers and an inequality concerning minimal graphs with a prescribed number of spanning trees”

Cyclic decompositions of complete graphs into spanning trees

Dalibor Froncek (2004)

Discussiones Mathematicae Graph Theory

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We examine decompositions of complete graphs with an even number of vertices, K 2 n , into n isomorphic spanning trees. While methods of such decompositions into symmetric trees have been known, we develop here a more general method based on a new type of vertex labelling, called flexible q-labelling. This labelling is a generalization of labellings introduced by Rosa and Eldergill.

Spanning caterpillars with bounded diameter

Ralph Faudree, Ronald Gould, Michael Jacobson, Linda Lesniak (1995)

Discussiones Mathematicae Graph Theory

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A caterpillar is a tree with the property that the vertices of degree at least 2 induce a path. We show that for every graph G of order n, either G or G̅ has a spanning caterpillar of diameter at most 2 log n. Furthermore, we show that if G is a graph of diameter 2 (diameter 3), then G contains a spanning caterpillar of diameter at most c n 3 / 4 (at most n).

On extremal sizes of locally k -tree graphs

Mieczysław Borowiecki, Piotr Borowiecki, Elżbieta Sidorowicz, Zdzisław Skupień (2010)

Czechoslovak Mathematical Journal

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A graph G is a if for any vertex v the subgraph induced by the neighbours of v is a k -tree, k ≥ 0 , where 0 -tree is an edgeless graph, 1 -tree is a tree. We characterize the minimum-size locally k -trees with n vertices. The minimum-size connected locally k -trees are simply ( k + 1 ) -trees. For k ≥ 1 , we construct locally k -trees which are maximal with respect to the spanning subgraph relation. Consequently, the number of edges in an n -vertex locally k -tree graph is between Ω ( n ) and O ( n 2 ) , where both bounds...

Multi-faithful spanning trees of infinite graphs

Norbert Polat (2001)

Czechoslovak Mathematical Journal

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For an end τ and a tree T of a graph G we denote respectively by m ( τ ) and m T ( τ ) the maximum numbers of pairwise disjoint rays of G and T belonging to τ , and we define t m ( τ ) : = min { m T ( τ ) T is a spanning tree of G } . In this paper we give partial answers—affirmative and negative ones—to the general problem of determining if, for a function f mapping every end τ of G to a cardinal f ( τ ) such that t m ( τ ) ≤ f ( τ ) ≤ m ( τ ) , there exists a spanning tree T of G such that m T ( τ ) = f ( τ ) for every end τ of G .

Minimum degree, leaf number and traceability

Simon Mukwembi (2013)

Czechoslovak Mathematical Journal

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Let G be a finite connected graph with minimum degree δ . The leaf number L ( G ) of G is defined as the maximum number of leaf vertices contained in a spanning tree of G . We prove that if δ ≥ 1 2 ( L ( G ) + 1 ) , then G is 2-connected. Further, we deduce, for graphs of girth greater than 4, that if δ ≥ 1 2 ( L ( G ) + 1 ) , then G contains a spanning path. This provides a partial solution to a conjecture of the computer program Graffiti.pc [DeLaVi na and Waller, Spanning trees with many leaves and average distance, Electron. J. Combin....