On the Shannon Capacity of Triangular Graphs

The Shannon capacity of a graph $G$ is $c(G)=\sup_{d\geq 1}(\alpha(G^d))^{\frac{1}{d}},$ where $\alpha(G)$ is the independence number of $G$. The Shannon capacity of the Kneser graph $\mathrm{KG}_{n,r}$ was determined by Lovász in 1979, but little is known about the Shannon capacity of the complement of that graph when $r$ does not divide $n$. The complement of the Kneser graph, $\overline{\mathrm{KG}}_{n,2}$, is also called the triangular graph $T_n$. The graph $T_n$ has the $n$-cycle $C_n$ as an induced subgraph, whereby $c(T_n) \geq c(C_n)$, and these two families of graphs are closely related in the current context as both can be considered via geometric packings of the discrete $d$-dimensional torus of width $n$ using two types of $d$-dimensional cubes of width $2$. Bounds on $c(T_n)$ obtained in this work include $c(T_7) \geq \sqrt[3]{35} \approx 3.271$, $c(T_{13}) \geq \sqrt[3]{248} \approx 6.283$, $c(T_{15}) \geq \sqrt[4]{2802} \approx 7.276$, and $c(T_{21}) \geq \sqrt[4]{11441} \approx 10.342$.

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Inductive graph invariants and approximation algorithms

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830922500197 ◽

2021 ◽

Author(s):

C. R. Subramanian

Keyword(s):

Optimization Problems ◽

Independence Number ◽

Optimal Solution ◽

Maximum Size ◽

Chordal Graphs ◽

Minimum Size ◽

Optimal Size ◽

Induced Subgraph ◽

Special Cases ◽

Optimal Formula

We introduce and study an inductively defined analogue [Formula: see text] of any increasing graph invariant [Formula: see text]. An invariant [Formula: see text] is increasing if [Formula: see text] whenever [Formula: see text] is an induced subgraph of [Formula: see text]. This inductive analogue simultaneously generalizes and unifies known notions like degeneracy, inductive independence number, etc., into a single generic notion. For any given increasing [Formula: see text], this gets us several new invariants and many of which are also increasing. It is also shown that [Formula: see text] is the minimum (over all orderings) of a value associated with each ordering. We also explore the possibility of computing [Formula: see text] (and a corresponding optimal vertex ordering) and identify some pairs [Formula: see text] for which [Formula: see text] can be computed efficiently for members of [Formula: see text]. In particular, it includes graphs of bounded [Formula: see text] values. Some specific examples (like the class of chordal graphs) have already been studied extensively. We further extend this new notion by (i) allowing vertex weighted graphs, (ii) allowing [Formula: see text] to take values from a totally ordered universe with a minimum and (iii) allowing the consideration of [Formula: see text]-neighborhoods for arbitrary but fixed [Formula: see text]. Such a generalization is employed in designing efficient approximations of some graph optimization problems. Precisely, we obtain efficient algorithms (by generalizing the known algorithm of Ye and Borodin [Y. Ye and A. Borodin, Elimination graphs, ACM Trans. Algorithms 8(2) (2012) 1–23] for special cases) for approximating optimal weighted induced [Formula: see text]-subgraphs and optimal [Formula: see text]-colorings (for hereditary [Formula: see text]’s) within multiplicative factors of (essentially) [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] denotes the inductive analogue (as defined in this work) of optimal size of an unweighted induced [Formula: see text]-subgraph of the input and [Formula: see text] is the minimum size of a forbidden induced subgraph of [Formula: see text]. Our results generalize the previous result on efficiently approximating maximum independent sets and minimum colorings on graphs of bounded inductive independence number to optimal [Formula: see text]-subgraphs and [Formula: see text]-colorings for arbitrary hereditary classes [Formula: see text]. As a corollary, it is also shown that any maximal [Formula: see text]-subgraph approximates an optimal solution within a factor of [Formula: see text] for unweighted graphs, where [Formula: see text] is maximum size of any induced [Formula: see text]-subgraph in any local neighborhood [Formula: see text].

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On the Normalized Shannon Capacity of a Union

Combinatorics Probability Computing ◽

10.1017/s0963548316000055 ◽

2016 ◽

Vol 25 (5) ◽

pp. 766-767 ◽

Cited By ~ 5

Author(s):

PETER KEEVASH ◽

EOIN LONG

Keyword(s):

Independence Number ◽

Shannon Capacity ◽

Strong Product ◽

Formula Group ◽

Image Position ◽

Product Of Graphs

Let G1 × G2 denote the strong product of graphs G1 and G2, that is, the graph on V(G1) × V(G2) in which (u1, u2) and (v1, v2) are adjacent if for each i = 1, 2 we have ui = vi or uivi ∈ E(Gi). The Shannon capacity of G is c(G) = limn → ∞ α(Gn)1/n, where Gn denotes the n-fold strong power of G, and α(H) denotes the independence number of a graph H. The normalized Shannon capacity of G is $$C(G) = \ffrac {\log c(G)}{\log |V(G)|}.$$ Alon [1] asked whether for every ε < 0 there are graphs G and G′ satisfying C(G), C(G′) < ε but with C(G + G′) > 1 − ε. We show that the answer is no.

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Vertex Covering with Monochromatic Pieces of few Colours

The Electronic Journal of Combinatorics ◽

10.37236/7469 ◽

2018 ◽

Vol 25 (3) ◽

Author(s):

Marlo Eugster ◽

Frank Mousset

Keyword(s):

Chromatic Number ◽

Complete Graph ◽

Independence Number ◽

Vertex Cover ◽

Constant Factor ◽

Regular Graphs ◽

Kneser Graph ◽

Vertex Covering ◽

Vertex Graph ◽

Positive Integers

In 1995, Erdös and Gyárfás proved that in every $2$-colouring of the edges of $K_n$, there is a vertex cover by $2\sqrt{n}$ monochromatic paths of the same colour, which is optimal up to a constant factor. The main goal of this paper is to study the natural multi-colour generalization of this problem: given two positive integers $r,s$, what is the smallest number $pc_{r,s}(K_n)$ such that in every colouring of the edges of $K_n$ with $r$ colours, there exists a vertex cover of $K_n$ by $pc_{r,s}(K_n)$ monochromatic paths using altogether at most $s$ different colours?For fixed integers $r>s$ and as $n\to\infty$, we prove that $pc_{r,s}(K_n) = \Theta(n^{1/\chi})$, where $\chi=\max{\{1,2+2s-r\}}$ is the chromatic number of the Kneser graph $KG(r,r-s)$. More generally, if one replaces $K_n$ by an arbitrary $n$-vertex graph with fixed independence number $\alpha$, then we have $pc_{r,s}(G) = O(n^{1/\chi})$, where this time around $\chi$ is the chromatic number of the Kneser hypergraph $KG^{(\alpha+1)}(r,r-s)$. This result is tight in the sense that there exist graphs with independence number $\alpha$ for which $pc_{r,s}(G) = \Omega(n^{1/\chi})$. This is in sharp contrast to the case $r=s$, where it follows from a result of Sárközy (2012) that $pc_{r,r}(G)$ depends only on $r$ and $\alpha$, but not on the number of vertices.We obtain similar results for the situation where instead of using paths, one wants to cover a graph with bounded independence number by monochromatic cycles, or a complete graph by monochromatic $d$-regular graphs.

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On The Independence Number Of Some Strong Products Of Cycle-Powers

Foundations of Computing and Decision Sciences ◽

10.1515/fcds-2015-0009 ◽

2015 ◽

Vol 40 (2) ◽

pp. 133-141 ◽

Cited By ~ 1

Author(s):

Marcin Jurkiewicz ◽

Marek Kubale ◽

Krzysztof Ocetkiewicz

Keyword(s):

Lower Bounds ◽

Independence Number ◽

Computational Results ◽

Shannon Capacity ◽

Noisy Channels ◽

Running Time ◽

The Third ◽

Independence Numbers

Abstract In the paper we give some theoretical and computational results on the third strong power of cycle-powers, for example, we have found the independence numbers α((C102)√3) = 30 and α((C144)√3) = 14. A number of optimizations have been introduced to improve the running time of our exhaustive algorithm used to establish the independence number of the third strong power of cycle-powers. Moreover, our results establish new exact values and/or lower bounds on the Shannon capacity of noisy channels.

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TRANSFERENCE FOR THE ERDŐS–KO–RADO THEOREM

Forum of Mathematics Sigma ◽

10.1017/fms.2015.21 ◽

2015 ◽

Vol 3 ◽

Cited By ~ 6

Author(s):

JÓZSEF BALOGH ◽

BÉLA BOLLOBÁS ◽

BHARGAV P. NARAYANAN

Keyword(s):

Random Graph ◽

Independence Number ◽

Independent Set ◽

Independent Interest ◽

Natural Numbers ◽

Kneser Graph ◽

Intersecting Family ◽

The Family

For natural numbers $n,r\in \mathbb{N}$ with $n\geqslant r$, the Kneser graph $K(n,r)$ is the graph on the family of $r$-element subsets of $\{1,\ldots ,n\}$ in which two sets are adjacent if and only if they are disjoint. Delete the edges of $K(n,r)$ with some probability, independently of each other: is the independence number of this random graph equal to the independence number of the Kneser graph itself? We shall answer this question affirmatively as long as $r/n$ is bounded away from $1/2$, even when the probability of retaining an edge of the Kneser graph is quite small. This gives us a random analogue of the Erdős–Ko–Rado theorem, since an independent set in the Kneser graph is the same as a uniform intersecting family. To prove our main result, we give some new estimates for the number of disjoint pairs in a family in terms of its distance from an intersecting family; these might be of independent interest.

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Independent Sets in the Union of Two Hamiltonian Cycles

The Electronic Journal of Combinatorics ◽

10.37236/6493 ◽

2018 ◽

Vol 25 (4) ◽

Author(s):

Ron Aharoni ◽

Daniel Soltész

Keyword(s):

Lower Bound ◽

Linear Function ◽

Independence Number ◽

Independent Set ◽

Hamiltonian Cycles ◽

Kneser Graph ◽

Threshold Phenomenon ◽

Technical Lemma ◽

Vertex Disjoint ◽

Element Set

Motivated by a question on the maximal number of vertex disjoint Schrijver graphs in the Kneser graph, we investigate the following function, denoted by $f(n,k)$: the maximal number of Hamiltonian cycles on an $n$ element set, such that no two cycles share a common independent set of size more than $k$. We shall mainly be interested in the behavior of $f(n,k)$ when $k$ is a linear function of $n$, namely $k=cn$. We show a threshold phenomenon: there exists a constant $c_t$ such that for $c<c_t$, $f(n,cn)$ is bounded by a constant depending only on $c$ and not on $n$, and for $c_t <c$, $f(n,cn)$ is exponentially large in $n ~(n \to \infty)$. We prove that $0.26 < c_t < 0.36$, but the exact value of $c_t$ is not determined. For the lower bound we prove a technical lemma, which for graphs that are the union of two Hamiltonian cycles establishes a relation between the independence number and the number of $K_4$ subgraphs. A corollary of this lemma is that if a graph $G$ on $n>12$ vertices is the union of two Hamiltonian cycles and $\alpha(G)=n/4$, then $V(G)$ can be covered by vertex-disjoint $K_4$ subgraphs.

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Turán-Ramsey Theorems and Kp-Independence Numbers

Combinatorics Probability Computing ◽

10.1017/s0963548300001218 ◽

1994 ◽

Vol 3 (3) ◽

pp. 297-325 ◽

Cited By ~ 10

Author(s):

P. Erdős ◽

A. Hajnal ◽

M. Simonovits ◽

V. T. Sós ◽

E. Szemerédi

Keyword(s):

Simple Structure ◽

Independence Number ◽

Independent Set ◽

Maximum Size ◽

Maximum Order ◽

Induced Subgraph ◽

Asymptotic Value ◽

Image Position ◽

Independence Numbers ◽

Edge Colouring

Let the Kp-independence number αp (G) of a graph G be the maximum order of an induced subgraph in G that contains no Kp. (So K2-independence number is just the maximum size of an independent set.) For given integers r, p, m > 0 and graphs L1,…,Lr, we define the corresponding Turán-Ramsey function RTp(n, L1,…,Lr, m) to be the maximum number of edges in a graph Gn of order n such that αp(Gn) ≤ m and there is an edge-colouring of G with r colours such that the jth colour class contains no copy of Lj, for j = 1,…, r. In this continuation of [11] and [12], we will investigate the problem where, instead of α(Gn) = o(n), we assume (for some fixed p > 2) the stronger condition that αp(Gn) = o(n). The first part of the paper contains multicoloured Turán-Ramsey theorems for graphs Gn of order n with small Kp-independence number αp(Gn). Some structure theorems are given for the case αp(Gn) = o(n), showing that there are graphs with fairly simple structure that are within o(n2) of the extremal size; the structure is described in terms of the edge densities between certain sets of vertices.The second part of the paper is devoted to the case r = 1, i.e., to the problem of determining the asymptotic value offor p < q. Several results are proved, and some other problems and conjectures are stated.

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On the Stability of Independence Polynomials

The Electronic Journal of Combinatorics ◽

10.37236/7280 ◽

2018 ◽

Vol 25 (1) ◽

Cited By ~ 1

Author(s):

Jason I. Brown ◽

Ben Cameron

Keyword(s):

Half Plane ◽

Complex Analysis ◽

Independence Number ◽

Independent Sets ◽

Induced Subgraph ◽

Left Half ◽

Independence Polynomial ◽

Independence Polynomials ◽

The Stability ◽

The Right

The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size and its roots are called independence roots. We investigate the stability of such polynomials, that is, conditions under which the independence roots lie in the left half-plane. We use results from complex analysis to determine graph operations that result in a stable or nonstable independence polynomial. In particular, we prove that every graph is an induced subgraph of a graph with stable independence polynomial. We also show that the independence polynomials of graphs with independence number at most three are necessarily stable, but for larger independence number, we show that the independence polynomials can have roots arbitrarily far to the right.

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