scholarly journals Spanning trees in graphs of minimum degree 4 or 5

1992 ◽  
Vol 104 (2) ◽  
pp. 167-183 ◽  
Author(s):  
Jerrold R. Griggs ◽  
Mingshen Wu
Keyword(s):  
Spanning Trees ◽  
Minimum Degree

scholarly journals Spanning Trees in Dense Graphs

2001 ◽  
Vol 10 (5) ◽  
pp. 397-416 ◽  
Author(s):  
JÁNOS KOMLÓS ◽  
GÁBOR N. SÁRKÓZY ◽  
ENDRE SZEMERÉDI
Keyword(s):  
Maximum Degree ◽  
Spanning Trees ◽  
Minimum Degree ◽  
Dense Graphs

In this paper we prove the following almost optimal theorem. For any δ > 0, there exist constants c and n0 such that, if n [ges ] n0, T is a tree of order n and maximum degree at most cn/log n, and G is a graph of order n and minimum degree at least (1/2 + δ)n, then T is a subgraph of G.

scholarly journals Spanning Trees with At Most 6 Leaves in K1,5-Free Graphs

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Pei Sun ◽  
Kai Liu
Keyword(s):  
Spanning Trees ◽  
Minimum Degree ◽  
Free Graph ◽  
Induced Subgraph ◽  
Degree Sum

A graph G is called K1,5-free if G contains no K1,5 as an induced subgraph. A tree with at most m leaves is called an m-ended tree. Let σkG be the minimum degree sum of k independent vertices in G. In this paper, it is shown that every connected K1,5-free graph G contains a spanning 6-ended tree if σ7G≥G−2.

scholarly journals Kocay's Lemma, Whitney's Theorem, and some Polynomial Invariant Reconstruction Problems

10.37236/1960 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Bhalchandra D. Thatte
Keyword(s):  
Characteristic Polynomial ◽  
Spanning Trees ◽  
Minimum Degree ◽  
Direct Consequence ◽  
Hamiltonian Cycles ◽  
Polynomial Invariant ◽  
Induced Subgraphs ◽  
Expansion Theorem ◽  
Rank Polynomial ◽  
Number Of Spanning Trees

Given a graph $G$, an incidence matrix ${\cal N}(G)$ is defined on the set of distinct isomorphism types of induced subgraphs of $G$. It is proved that Ulam's conjecture is true if and only if the ${\cal N}$-matrix is a complete graph invariant. Several invariants of a graph are then shown to be reconstructible from its ${\cal N}$-matrix. The invariants include the characteristic polynomial, the rank polynomial, the number of spanning trees and the number of hamiltonian cycles in a graph. These results are stronger than the original results of Tutte in the sense that actual subgraphs are not used. It is also proved that the characteristic polynomial of a graph with minimum degree 1 can be computed from the characteristic polynomials of all its induced proper subgraphs. The ideas in Kocay's lemma play a crucial role in most proofs. Kocay's lemma is used to prove Whitney's subgraph expansion theorem in a simple manner. The reconstructibility of the characteristic polynomial is then demonstrated as a direct consequence of Whitney's theorem as formulated here.

scholarly journals Distributed Minimum Degree Spanning Trees

2019 ◽  
Author(s):  
Michael Dinitz ◽  
Magnus M. Halldorsson ◽  
Taisuke Izumi ◽  
Calvin Newport
Keyword(s):  
Spanning Trees ◽  
Minimum Degree
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