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 Total Domination and Matching Numbers in Claw-Free Graphs

10.37236/1085 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Anders Yeo
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Matching Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Free Graph ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Total Dominating Set

A set $M$ of edges of a graph $G$ is a matching if no two edges in $M$ are incident to the same vertex. The matching number of $G$ is the maximum cardinality of a matching of $G$. A set $S$ of vertices in $G$ is a total dominating set of $G$ if every vertex of $G$ is adjacent to some vertex in $S$. The minimum cardinality of a total dominating set of $G$ is the total domination number of $G$. If $G$ does not contain $K_{1,3}$ as an induced subgraph, then $G$ is said to be claw-free. We observe that the total domination number of every claw-free graph with minimum degree at least three is bounded above by its matching number. In this paper, we use transversals in hypergraphs to characterize connected claw-free graphs with minimum degree at least three that have equal total domination and matching numbers.

scholarly journals Separation Choosability and Dense Bipartite Induced Subgraphs

2019 ◽  
Vol 28 (5) ◽  
pp. 720-732 ◽  
Author(s):  
Louis Esperet ◽  
Ross J. Kang ◽  
Stéphan Thomassé
Keyword(s):  
Minimum Degree ◽  
Bipartite Graphs ◽  
Independent Interest ◽  
List Size ◽  
Induced Subgraphs ◽  
Natural Class ◽  
Free Graph ◽  
Induced Subgraph ◽  
Restricted Form ◽  
Ramsey Type

AbstractWe study a restricted form of list colouring, for which every pair of lists that correspond to adjacent vertices may not share more than one colour. The optimal list size such that a proper list colouring is always possible given this restriction, we call separation choosability. We show for bipartite graphs that separation choosability increases with (the logarithm of) the minimum degree. This strengthens results of Molloy and Thron and, partially, of Alon. One attempt to drop the bipartiteness assumption precipitates a natural class of Ramsey-type questions, of independent interest. For example, does every triangle-free graph of minimum degree d contain a bipartite induced subgraph of minimum degree Ω(log d) as d→∞?

scholarly journals Bipartite Induced Density in Triangle-Free Graphs

10.37236/8650 ◽  
2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Wouter Cames van Batenburg ◽  
Rémi De Joannis de Verclos ◽  
Ross J. Kang ◽  
François Pirot
Keyword(s):  
Chromatic Number ◽  
Minimum Degree ◽  
Free Graph ◽  
Induced Subgraph ◽  
Logarithmic Factor ◽  
Fractional Chromatic Number ◽  
List Chromatic Number

We prove that any triangle-free graph on $n$ vertices with minimum degree at least $d$ contains a bipartite induced subgraph of minimum degree at least $d^2/(2n)$. This is sharp up to a logarithmic factor in $n$. Relatedly, we show that the fractional chromatic number of any such triangle-free graph is at most the minimum of $n/d$ and $(2+o(1))\sqrt{n/\log n}$ as $n\to\infty$. This is sharp up to constant factors. Similarly, we show that the list chromatic number of any such triangle-free graph is at most $O(\min\{\sqrt{n},(n\log n)/d\})$ as $n\to\infty$. Relatedly, we also make two conjectures. First, any triangle-free graph on $n$ vertices has fractional chromatic number at most $(\sqrt{2}+o(1))\sqrt{n/\log n}$ as $n\to\infty$. Second, any  triangle-free graph on $n$ vertices has list chromatic number at most $O(\sqrt{n/\log n})$ as $n\to\infty$.

scholarly journals Toughness, Forbidden Subgraphs and Pancyclicity

2021 ◽  
Vol 37 (3) ◽  
pp. 839-866
Author(s):  
Wei Zheng ◽  
Hajo Broersma ◽  
Ligong Wang
Keyword(s):  
Recent Article ◽  
Forbidden Subgraph ◽  
Forbidden Subgraphs ◽  
Free Graph ◽  
Induced Subgraph ◽  
Complete Answer ◽  
Open Case

AbstractMotivated by several conjectures due to Nikoghosyan, in a recent article due to Li et al., the aim was to characterize all possible graphs H such that every 1-tough H-free graph is hamiltonian. The almost complete answer was given there by the conclusion that every proper induced subgraph H of $$K_1\cup P_4$$ K 1 ∪ P 4 can act as a forbidden subgraph to ensure that every 1-tough H-free graph is hamiltonian, and that there is no other forbidden subgraph with this property, except possibly for the graph $$K_1\cup P_4$$ K 1 ∪ P 4 itself. The hamiltonicity of 1-tough $$K_1\cup P_4$$ K 1 ∪ P 4 -free graphs, as conjectured by Nikoghosyan, was left there as an open case. In this paper, we consider the stronger property of pancyclicity under the same condition. We find that the results are completely analogous to the hamiltonian case: every graph H such that any 1-tough H-free graph is hamiltonian also ensures that every 1-tough H-free graph is pancyclic, except for a few specific classes of graphs. Moreover, there is no other forbidden subgraph having this property. With respect to the open case for hamiltonicity of 1-tough $$K_1\cup P_4$$ K 1 ∪ P 4 -free graphs we give infinite families of graphs that are not pancyclic.

scholarly journals Another Infinite Sequence of Dense Triangle-Free Graphs

10.37236/1381 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Stephan Brandt ◽  
Tomaž Pisanski
Keyword(s):  
Structural Properties ◽  
Chromatic Number ◽  
Infinite Sequence ◽  
Minimum Degree ◽  
Free Graph ◽  
The Core ◽  
Graph Homomorphisms ◽  
Natural Way

The core is the unique homorphically minimal subgraph of a graph. A triangle-free graph with minimum degree $\delta > n/3$ is called dense. It was observed by many authors that dense triangle-free graphs share strong structural properties and that the natural way to describe the structure of these graphs is in terms of graph homomorphisms. One infinite sequence of cores of dense maximal triangle-free graphs was known. All graphs in this sequence are 3-colourable. Only two additional cores with chromatic number 4 were known. We show that the additional graphs are the initial terms of a second infinite sequence of cores.

Color degree sum conditions for properly colored spanning trees in edge-colored graphs

2020 ◽  
Vol 343 (11) ◽  
pp. 112042
Author(s):  
Mikio Kano ◽  
Shun-ichi Maezawa ◽  
Katsuhiro Ota ◽  
Masao Tsugaki ◽  
Takamasa Yashima
Keyword(s):  
Spanning Trees ◽  
Degree Sum ◽  
Colored Graphs ◽  
Color Degree

scholarly journals Thek-Core and Branching Processes

2008 ◽  
Vol 17 (1) ◽  
pp. 111-136 ◽  
Author(s):  
OLIVER RIORDAN
Keyword(s):  
Power Law ◽  
Branching Process ◽  
Minimum Degree ◽  
Branching Processes ◽  
Threshold Value ◽  
Free Graph ◽  
Scale Free ◽  
Asymptotic Size ◽  
Inhomogeneous Random Graphs ◽  
Local Coupling

Thek-coreof a graphGis the maximal subgraph ofGhaving minimum degree at leastk. In 1996, Pittel, Spencer and Wormald found the threshold λcfor the emergence of a non-trivialk-core in the random graphG(n, λ/n), and the asymptotic size of thek-core above the threshold. We give a new proof of this result using a local coupling of the graph to a suitable branching process. This proof extends to a general model of inhomogeneous random graphs with independence between the edges. As an example, we study thek-core in a certain power-law or ‘scale-free’ graph with a parameterccontrolling the overall density of edges. For eachk≥ 3, we find the threshold value ofcat which thek-core emerges, and the fraction of vertices in thek-core whencis ϵ above the threshold. In contrast toG(n, λ/n), this fraction tends to 0 as ϵ→0.

Rainbow connection numbers and the minimum degree sum of a graph

2013 ◽  
Vol 43 (1) ◽  
pp. 7-14 ◽  
Author(s):  
XueLiang LI ◽  
JiuYing DONG
Keyword(s):  
Minimum Degree ◽  
Degree Sum ◽  
Rainbow Connection

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
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