On the Maximum Number of Triangles in Wheel-Free Graphs

Author(s):

Zoltán Füredi ◽

Michel X. Goemans ◽

Daniel J. Kleitman

Keyword(s):

Upper Bound ◽

Gallai [1] raised the question of determining t(n), the maximum number of triangles in graphs of n vertices with acyclic neighborhoods. Here we disprove his conjecture (t(n) ~ n2/8) by exhibiting graphs having n2/7.5 triangles. We improve the upper bound [11] of (n2 − n)/6 to t(n) ≤; n2/7.02 + O(n). For regular graphs, we further decrease this bound to n2/7.75 + O(n).

Cubic Graphs with Small Independence Ratio

10.37236/7272 ◽

2019 ◽

Vol 26 (1) ◽

Author(s):

József Balogh ◽

Alexandr Kostochka ◽

Xujun Liu

Keyword(s):

Lower Bounds ◽

Upper Bound ◽

Independence Number ◽

Regular Graphs ◽

Cubic Graphs

Let $i(r,g)$ denote the infimum of the ratio $\frac{\alpha(G)}{|V(G)|}$ over the $r$-regular graphs of girth at least $g$, where $\alpha(G)$ is the independence number of $G$, and let $i(r,\infty) := \lim\limits_{g \to \infty} i(r,g)$. Recently, several new lower bounds of $i(3,\infty)$ were obtained. In particular, Hoppen and Wormald showed in 2015 that $i(3, \infty) \geqslant 0.4375,$ and Csóka improved it to $i(3,\infty) \geqslant 0.44533$ in 2016. Bollobás proved the upper bound $i(3,\infty) < \frac{6}{13}$ in 1981, and McKay improved it to $i(3,\infty) < 0.45537$in 1987. There were no improvements since then. In this paper, we improve the upper bound to $i(3,\infty) \leqslant 0.454.$

On the Widom–Rowlinson Occupancy Fraction in Regular Graphs

Combinatorics Probability Computing ◽

10.1017/s0963548316000249 ◽

2016 ◽

Vol 26 (2) ◽

pp. 183-194 ◽

Cited By ~ 4

Author(s):

EMMA COHEN ◽

WILL PERKINS ◽

PRASAD TETALI

Keyword(s):

Partition Function ◽

Regular Graph ◽

Upper Bound ◽

Complete Graphs ◽

Regular Graphs ◽

Interacting Particles ◽

Special Case ◽

Random Configuration

We consider the Widom–Rowlinson model of two types of interacting particles on d-regular graphs. We prove a tight upper bound on the occupancy fraction, the expected fraction of vertices occupied by a particle under a random configuration from the model. The upper bound is achieved uniquely by unions of complete graphs on d + 1 vertices, Kd+1. As a corollary we find that Kd+1 also maximizes the normalized partition function of the Widom–Rowlinson model over the class of d-regular graphs. A special case of this shows that the normalized number of homomorphisms from any d-regular graph G to the graph HWR, a path on three vertices with a loop on each vertex, is maximized by Kd+1. This proves a conjecture of Galvin.

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 ◽

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.

Identifying Vertex Covers in Graphs

10.37236/2114 ◽

2012 ◽

Vol 19 (4) ◽

Cited By ~ 2

Author(s):

Michael A Henning ◽

Anders Yeo

Keyword(s):

Regular Graph ◽

Upper Bound ◽

Maximum Degree ◽

Vertex Cover ◽

Nonempty Intersection ◽

Petersen Graph ◽

Minimum Size ◽

Regular Graphs ◽

Packing Number

An identifying vertex cover in a graph $G$ is a subset $T$ of vertices in $G$ that has a nonempty intersection with every edge of $G$ such that $T$ distinguishes the edges, that is, $e \cap T \ne \emptyset$ for every edge $e$ in $G$ and $e \cap T \ne f \cap T$ for every two distinct edges $e$ and $f$ in $G$. The identifying vertex cover number $\tau_D(G)$ of $G$ is the minimum size of an identifying vertex cover in $G$. We observe that $\tau_D(G) + \rho(G) = |V(G)|$, where $\rho(G)$ denotes the packing number of $G$. We conjecture that if $G$ is a graph of order $n$ and size $m$ with maximum degree $\Delta$, then $\tau_D(G) \le \left( \frac{\Delta(\Delta - 1)}{\Delta^2 + 1} \right) n + \left( \frac{2}{\Delta^2 + 1} \right) m$. If the conjecture is true, then the bound is best possible for all $\Delta \ge 1$. We prove this conjecture when $\Delta \ge 1$ and $G$ is a $\Delta$-regular graph. The three known Moore graphs of diameter two, namely the $5$-cycle, the Petersen graph and the Hoffman-Singleton graph, are examples of regular graphs that achieves equality in the upper bound. We also prove this conjecture when $\Delta \in \{2,3\}$.

An upper bound for the number of independent sets in regular graphs

Discrete Mathematics ◽

10.1016/j.disc.2009.06.031 ◽

2009 ◽

Vol 309 (23-24) ◽

pp. 6635-6640 ◽

Cited By ~ 5

Author(s):

David Galvin

Keyword(s):

Upper Bound ◽

Independent Sets ◽

An upper bound for domination number of 5-regular graphs

Czechoslovak Mathematical Journal ◽

10.1007/s10587-006-0079-4 ◽

2006 ◽

Vol 56 (3) ◽

pp. 1049-1061 ◽

Author(s):

Hua-Ming Xing ◽

Liang Sun ◽

Xue-Gang Chen

Keyword(s):

Upper Bound ◽

Domination Number ◽

On Vertex Covering Transversal Domination Number of Regular Graphs

The Scientific World JOURNAL ◽

10.1155/2016/1029024 ◽

2016 ◽

Vol 2016 ◽

pp. 1-7 ◽

Author(s):

R. Vasanthi ◽

K. Subramanian

Keyword(s):

Upper Bound ◽

Domination Number ◽

Simple Graph ◽

Regular Graphs ◽

Dominating Sets ◽

Cubic Graph ◽

Minimum Cardinality ◽

Vertex Covering

A simple graphG=(V,E)is said to ber-regular if each vertex ofGis of degreer. The vertex covering transversal domination numberγvct(G)is the minimum cardinality among all vertex covering transversal dominating sets ofG. In this paper, we analyse this parameter on different kinds of regular graphs especially forQnandH3,n. Also we provide an upper bound forγvctof a connected cubic graph of ordern≥8. Then we try to provide a more stronger relationship betweenγandγvct.

A Generalized Information-Theoretic Approach for Bounding the Number of Independent Sets in Bipartite Graphs

Entropy ◽

10.3390/e23030270 ◽

2021 ◽

Vol 23 (3) ◽

pp. 270

Author(s):

Igal Sason

Keyword(s):

Upper Bound ◽

Independent Sets ◽

Bipartite Graphs ◽

Theoretic Approach ◽

Regular Graphs ◽

Proof Technique ◽

Information Theoretic ◽

Theoretic Proof ◽

Information Theoretic Approach ◽

General Graphs

This paper studies the problem of upper bounding the number of independent sets in a graph, expressed in terms of its degree distribution. For bipartite regular graphs, Kahn (2001) established a tight upper bound using an information-theoretic approach, and he also conjectured an upper bound for general graphs. His conjectured bound was recently proved by Sah et al. (2019), using different techniques not involving information theory. The main contribution of this work is the extension of Kahn’s information-theoretic proof technique to handle irregular bipartite graphs. In particular, when the bipartite graph is regular on one side, but may be irregular on the other, the extended entropy-based proof technique yields the same bound as was conjectured by Kahn (2001) and proved by Sah et al. (2019).

Some Applications of the Proper and Adjacency Polynomials in the Theory of Graph Spectra

10.37236/1306 ◽

1997 ◽

Vol 4 (1) ◽

Author(s):

M. A. Fiol

Keyword(s):

Orthogonal Polynomials ◽

Regular Graph ◽

Upper Bound ◽

Association Schemes ◽

Regular Graphs ◽

Local Spectrum ◽

Graph Spectra ◽

Distance Regular Graphs ◽

New Applications

Given a vertex $u\in V$ of a graph $G=(V,E)$, the (local) proper polynomials constitute a sequence of orthogonal polynomials, constructed from the so-called $u$-local spectrum of $G$. These polynomials can be thought of as a generalization, for all graphs, of the distance polynomials for the distance-regular graphs. The (local) adjacency polynomials, which are basically sums of proper polynomials, were recently used to study a new concept of distance-regularity for non-regular graphs, and also to give bounds on some distance-related parameters such as the diameter. Here some new applications of both, the proper and adjacency polynomials, are derived, such as bounds for the radius of $G$ and the weight $k$-excess of a vertex. Given the integers $k,\mu\geq 0$, let $G_k^\mu(u)$ denote the set of vertices which are at distance at least $k$ from a vertex $u\in V$, and there exist exactly $\mu$ (shortest) $k$-paths from $u$ to each of such vertices. As a main result, an upper bound for the cardinality of $G_k^\mu(u)$ is derived, showing that $|G_k^\mu(u)|$ decreases at least as $O(\mu^{-2})$, and the cases in which the bound is attained are characterized. When these results are particularized to regular graphs with four distinct eigenvalues, we reobtain a result of Van Dam about 3-class association schemes, and prove some conjectures of Haemers and Van Dam, about the number of vertices at distance three from every vertex of a regular graph with four distinct eigenvalues —setting $k=2$ and $\mu=0$— and, more generally, the number of non-adjacent vertices to every vertex $u\in V$, which have $\mu$ common neighbours with it.

Rate of Convergence of the Short Cycle Distribution in Random Regular Graphs Generated by Pegging

10.37236/133 ◽

2009 ◽

Vol 16 (1) ◽

Author(s):

Pu Gao ◽

Nicholas Wormald

Keyword(s):

Rate Of Convergence ◽

Total Variation ◽

Upper Bound ◽

Mixing Time ◽

Limiting Distribution ◽

Total Variation Distance ◽

Regular Graphs ◽

Short Cycle ◽

Variation Distance

The pegging algorithm is a method of generating large random regular graphs beginning with small ones. The $\epsilon$-mixing time of the distribution of short cycle counts of these random regular graphs is the time at which the distribution reaches and maintains total variation distance at most $\epsilon$ from its limiting distribution. We show that this $\epsilon$-mixing time is not $o(\epsilon^{-1})$. This demonstrates that the upper bound $O(\epsilon^{-1})$ proved recently by the authors is essentially tight.