scholarly journals A Degree Sequence Version of the Kühn–Osthus Tiling Theorem

10.37236/8986 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Joseph Hyde ◽  
Andrew Treglown
Keyword(s):  
Minimum Degree ◽  
Degree Sequence ◽  
Lower Degree ◽  
Fundamental Result ◽  
Additive Constant ◽  
Perfect Graph

A fundamental result of Kühn and Osthus [The minimum degree threshold for perfect graph packings, Combinatorica, 2009] determines up to an additive constant the minimum degree threshold that forces a graph to contain a perfect $H$-tiling. We prove a degree sequence version of this result which allows for a significant number of vertices to have lower degree.

scholarly journals The Diameter of Almost Eulerian Digraphs

10.37236/429 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Peter Dankelmann ◽  
L. Volkmann
Keyword(s):  
Graph Theory ◽  
Directed Graph ◽  
Upper Bound ◽  
Minimum Degree ◽  
Additive Constant ◽  
Undirected Graphs

Soares [J. Graph Theory 1992] showed that the well known upper bound $\frac{3}{\delta+1}n+O(1)$ on the diameter of undirected graphs of order $n$ and minimum degree $\delta$ also holds for digraphs, provided they are eulerian. In this paper we investigate if similar bounds can be given for digraphs that are, in some sense, close to being eulerian. In particular we show that a directed graph of order $n$ and minimum degree $\delta$ whose arc set can be partitioned into $s$ trails, where $s\leq \delta-2$, has diameter at most $3 ( \delta+1 - \frac{s}{3})^{-1}n+O(1)$. If $s$ also divides $\delta-2$, then we show the diameter to be at most $3(\delta+1 - \frac{(\delta-2)s}{3(\delta-2)+s} )^{-1}n+O(1)$. The latter bound is sharp, apart from an additive constant. As a corollary we obtain the sharp upper bound $3( \delta+1 - \frac{\delta-2}{3\delta-5})^{-1} n + O(1)$ on the diameter of digraphs that have an eulerian trail.

scholarly journals A new graph over semi-direct products of groups

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 611-619
Author(s):  
Sercan Topkaya ◽  
Sinan Cevik
Keyword(s):  
Chromatic Number ◽  
Final Section ◽  
Degree Sequence ◽  
Domination Number ◽  
Clique Number ◽  
Perfect Graph ◽  
Direct Products ◽  
Products Of Groups ◽  
Minimum Degrees ◽  
Product Of Groups

In this paper, by establishing a new graph ?(G) over the semi-direct product of groups, we will first state and prove some graph-theoretical properties, namely, diameter, maximum and minimum degrees, girth, degree sequence, domination number, chromatic number, clique number of ?(G). In the final section we will show that ?(G) is actually a perfect graph.

scholarly journals On Perfect Packings in Dense Graphs

10.37236/3173 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
József Balogh ◽  
Alexandr Kostochka ◽  
Andrew Treglown
Keyword(s):  
Minimum Degree ◽  
Degree Sequence ◽  
Edge Density ◽  
Structural Result ◽  
Dense Graphs ◽  
Delta G ◽  
Vertex Disjoint ◽  
Density Threshold

We say that a graph $G$ has a perfect $H$-packing if there exists a set of vertex-disjoint copies of $H$ which cover all the vertices in $G$. We consider various problems concerning perfect $H$-packings: Given $n, r , D \in \mathbb N$, we characterise the edge density threshold that ensures a perfect $K_r$-packing in any graph $G$ on $n$ vertices and with minimum degree $\delta (G) \geq D$. We also give two conjectures concerning degree sequence conditions which force a graph to contain a perfect $H$-packing. Other related embedding problems are also considered. Indeed, we give a structural result concerning $K_r$-free graphs that satisfy a certain degree sequence condition.

scholarly journals Distant Set Distinguishing Total Colourings of Graphs

10.37236/5481 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Jakub Przybyło
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Probabilistic Method ◽  
Constant Factor ◽  
Additive Constant ◽  
Regular Graphs

The Total Colouring Conjecture suggests that $\Delta+3$ colours ought to suffice in order to provide a proper total colouring of every graph $G$ with maximum degree $\Delta$. Thus far this has been confirmed up to an additive constant factor, and the same holds even if one additionally requires every pair of neighbours in $G$ to differ with respect to the sets of their incident colours, so called pallets. Within this paper we conjecture that an upper bound of the form $\Delta+C$, for a constant $C>0$ still remains valid even after extending the distinction requirement to pallets associated with vertices at distance at most $r$, if only $G$ has minimum degree $\delta$ larger than a constant dependent on $r$. We prove that such assumption on $\delta$ is then unavoidable and exploit the probabilistic method in order to provide two supporting results for the conjecture. Namely, we prove the upper bound $(1+o(1))\Delta$ for every $r$, and show that for any fixed $\epsilon\in(0,1]$ and $r$, the conjecture holds if $\delta\geq \varepsilon\Delta$, i.e., in particular for regular graphs.

scholarly journals On the structure of bidegreed graphs with minimal spectral radius

Filomat ◽  
2014 ◽  
Vol 28 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Francesco Belardo
Keyword(s):  
Spectral Radius ◽  
Adjacency Matrix ◽  
Limit Point ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Degree Sequence

A graph is said to be (?,?)-bidegreed if vertices all have one of two possible degrees: the maximum degree ? or the minimum degree ?, with ? ? ?. We show that in the set of connected (?,1)- bidegreed graphs, other than trees, with prescribed degree sequence, the graphs minimizing the adjacency matrix spectral radius cannot have vertices adjacent to ? - 1 vertices of degree 1, that is, there are not any hanging trees. Further we consider the limit point for the spectral radius of (?,1)-bidegreed graphs when degree ? vertices are inserted in each edge between any two degree ? vertices.

scholarly journals On degree-sequence characterization and the extremal number of edges for various Hamiltonian properties under fault tolerance

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Shih-Yan Chen ◽  
Shin-Shin Kao ◽  
Hsun Su
Keyword(s):  
Minimum Degree ◽  
Degree Sequence ◽  
Sequence Characterization ◽  
Hamiltonian Connected ◽  
Vertex Graph ◽  
Minimum Number ◽  
Hamiltonian Connectivity ◽  
International Audience ◽  
Delta G ◽  
Hamiltonian Properties

International audience Assume that $n, \delta ,k$ are integers with $0 \leq k < \delta < n$. Given a graph $G=(V,E)$ with $|V|=n$. The symbol $G-F, F \subseteq V$, denotes the graph with $V(G-F)=V-F$, and $E(G-F)$ obtained by $E$ after deleting the edges with at least one endvertex in $F$. $G$ is called <i>$k$-vertex fault traceable</i>, <i>$k$-vertex fault Hamiltonian</i>, or <i>$k$-vertex fault Hamiltonian-connected</i> if $G-F$ remains traceable, Hamiltonian, and Hamiltonian-connected for all $F$ with $0 \leq |F| \leq k$, respectively. The notations $h_1(n, \delta ,k)$, $h_2(n, \delta ,k)$, and $h_3(n, \delta ,k)$ denote the minimum number of edges required to guarantee an $n$-vertex graph with minimum degree $\delta (G) \geq \delta$ to be $k$-vertex fault traceable, $k$-vertex fault Hamiltonian, and $k$-vertex fault Hamiltonian-connected, respectively. In this paper, we establish a theorem which uses the degree sequence of a given graph to characterize the $k$-vertex fault traceability/hamiltonicity/Hamiltonian-connectivity, respectively. Then we use this theorem to obtain the formulas for $h_i(n, \delta ,k)$ for $1 \leq i \leq 3$, which improves and extends the known results for $k=0$.

scholarly journals Embedding Spanning Bipartite Graphs of Small Bandwidth

2012 ◽  
Vol 22 (1) ◽  
pp. 71-96 ◽  
Author(s):  
FIACHRA KNOX ◽  
ANDREW TREGLOWN
Keyword(s):  
Bipartite Graph ◽  
Minimum Degree ◽  
Degree Sequence ◽  
Bipartite Graphs ◽  
Type Result ◽  
Bounded Degree ◽  
Expansion Property ◽  
Small Bandwidth ◽  
Bipartite Case

Böttcher, Schacht and Taraz (Math. Ann., 2009) gave a condition on the minimum degree of a graph G on n vertices that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth o(n), thereby proving a conjecture of Bollobás and Komlós (Combin. Probab. Comput., 1999). We strengthen this result in the case when H is bipartite. Indeed, we give an essentially best-possible condition on the degree sequence of a graph G on n vertices that forces G to contain every bipartite graph H on n vertices of bounded degree and of bandwidth o(n). This also implies an Ore-type result. In fact, we prove a much stronger result where the condition on G is relaxed to a certain robust expansion property. Our result also confirms the bipartite case of a conjecture of Balogh, Kostochka and Treglown concerning the degree sequence of a graph which forces a perfect H-packing.

scholarly journals Cover time of a random graph with given degree sequence

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Mohammed Abdullah ◽  
Colin Cooper ◽  
Alan Frieze
Keyword(s):  
Random Graph ◽  
High Probability ◽  
Minimum Degree ◽  
Degree Sequence ◽  
Average Degree ◽  
Cover Time ◽  
Vertex Set ◽  
International Audience

International audience In this paper we establish the cover time of a random graph $G(\textbf{d})$ chosen uniformly at random from the set of graphs with vertex set $[n]$ and degree sequence $\textbf{d}$. We show that under certain restrictions on $\textbf{d}$, the cover time of $G(\textbf{d})$ is with high probability asymptotic to $\frac{d-1}{ d-2} \frac{\theta}{ d}n \log n$. Here $\theta$ is the average degree and $d$ is the $\textit{effective minimum degree}$. The effective minimum degree is the first entry in the sorted degree sequence which occurs order $n$ times.

Radius, Diameter and the Degree Sequence of a Graph

2015 ◽  
Vol 65 (6) ◽  
Author(s):  
Jaya Percival Mazorodze ◽  
Simon Mukwembi
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Degree Sequence ◽  
Upper Bounds ◽  
Free Graph

AbstractWe give asymptotically sharp upper bounds on the radius and diameter of(i) a connected graph,(ii) a connected triangle-free graph,(iii) a connected C4-free graph of given order, minimum degree, and maximum degree.We also give better bounds on the radius and diameter for triangle-free graphs with a given order, minimum degree and a given number of distinct terms in the degree sequence of the graph. Our results improve on old classical theorems by Erd˝os, Pach, Pollack and Tuza [Radius, diameter, and minimum degree, J. Combin. Theory Ser. B 47 (1989), 73-79] on radius, diameter and minimum degree.

scholarly journals Counting Trees in Graphs

10.37236/6084 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Jacques Verstraete ◽  
Dhruv Mubayi
Keyword(s):  
Lower Bound ◽  
Convex Function ◽  
Disjoint Union ◽  
Minimum Degree ◽  
Degree Sequence ◽  
Average Degree ◽  
Vertex Graph ◽  
Counting Trees

Erdős and Simonovits proved that the number of paths of length $t$ in an $n$-vertex graph of average degree $d$ is at least $(1 - \delta) nd(d - 1) \cdots (d - t + 1)$, where $\delta = (\log d)^{-1/2 + o(1)}$ as $d \rightarrow \infty$. In this paper, we strengthen and generalize this result as follows. Let $T$ be a tree with $t$ edges. We prove that for any $n$-vertex graph $G$ of average degree $d$ and minimum degree greater than $t$, the number of labelled copies of $T$ in $G$ is at least \[(1 - \varepsilon) n d(d - 1) \cdots (d - t + 1)\] where $\varepsilon = O(d^{-2})$ as $d \rightarrow \infty$. This bound is tight except for the term $1 - \varepsilon$, as shown by a disjoint union of cliques. Our proof is obtained by first showing a lower bound that is a convex function of the degree sequence of $G$, and this answers a question of Dellamonica et. al.

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