Let $r\geq 3$ be an integer and $G$ be a graph. Let $δ(G), Δ(G)$, $α(G)$ and $μ(G)$ denotes minimum degree, maximum degree, independence number and matching number of $G$, respectively. Recently, Caro, Davila and Pepper proved $δ(G)α(G)\leq Δ(G)μ(G)$. Mohr and Rautenbach characterized the extremal graphs for non-regular graphs and 3-regular graphs. In this note, we characterize the extremal graphs for all $r$-regular graphs in term of Gallai-Edmonds Structure Theorem, which extends Mohr and Rautenbach's result.
We disprove the characterisation of graphs with equal independence and annihilation number by Larson and Pepper (2011). Series of counterexamples with arbitrary number of vertices, arbitrary number of components, arbitrary large independence number and arbitrary large difference between the critical and the regular independence number are provided. Furthermore, we point out the error in the proof of the theorem. However, we show that the theorem still holds for bipartite graphs and connected claw-free graphs.
Caro, Davila, and Pepper (arXiv:1909.09093) recently proved $δ(G) α(G)\leq Δ(G) μ(G)$ for every graph $G$ with minimum degree $δ(G)$, maximum degree $Δ(G)$, independence number $α(G)$, and matching number $μ(G)$. Answering some problems they posed, we characterize the extremal graphs for $δ(G)<Δ(G)$ as well as for $δ(G)=Δ(G)=3$.
In this article, we define a family of regular bipartite graphs and show that the homotopy type of the independence complexes of this family is the wedge sum of spheres of certain dimensions.
The \emph{regular independence number}, introduced by Albertson and Boutin in 1990, is the maximum cardinality of an independent set of $G$ in which all vertices have equal degree in $G$. Recently, Caro, Hansberg and Pepper introduced the concept of regular $k$-independence number, which is a natural generalization of the regular independence number. A \emph{$k$-independent set} is a set of vertices whose induced subgraph has maximum degree at most $k$. The \emph{regular $k$-independence number} of $G$, denoted by $α_{k-reg}(G)$, is defined as the maximum cardinality of a $k$-independent set o…
The annihilation number $a(G)$ of a graph $G$ is an efficiently computable upper bound on the independence number $α(G)$ of $G$. Recently, Hiller observed that a characterization of the graphs $G$ with $α(G)=a(G)$ due to Larson and Pepper is false. Since the known efficient algorithm recognizing these graphs was based on this characterization, the complexity of recognizing graphs $G$ with $α(G)=a(G)$ was once again open. We show that these graphs can indeed be recognized efficiently. More generally, we show that recognizing graphs $G$ with $α(G)\geq a(G)-\ell$ is fixed parameter tractable usin…
An independent set $I_c$ is a \textit{critical independent set} if $|I_c| - |N(I_c)| \geq |J| - |N(J)|$, for any independent set $J$. The \textit{critical independence number} of a graph is the cardinality of a maximum critical independent set. This number is a lower bound for the independence number and can be computed in polynomial-time. Any graph can be decomposed into two subgraphs where the independence number of one subgraph equals its critical independence number, where the critical independence number of the other subgraph is zero, and where the sum of the independence numbers of the s…
The independence polynomial $I(G, x)$ of a graph $G$ is the polynomial in variable $x$ in which the coefficient $a_n$ on $x^n$ gives the number of independent subsets $S \subseteq V(G)$ of vertices of $G$ such that $|S| = n$. $I(G, x)$ is unimodal if there is an index $μ$ such that that $a_0 \leq a_1 \leq$...$\leq a_{μ-1} \leq a_μ \geq a_{μ+1} \geq$...$\geq a_{d-1} \geq a_d$ While the independence polynomials of many families of graphs with highly regular structure are known to be unimodal, little is known about less regularly structured graphs. We analyze the independence polynomials of a lar…
The Grundy domination number, ${γ_{\rm gr}}(G)$, of a graph $G$ is the maximum length of a sequence $(v_1,v_2,\ldots, v_k)$ of vertices in $G$ such that for every $i\in \{2,\ldots, k\}$, the closed neighborhood $N[v_i]$ contains a vertex that does not belong to any closed neighborhood $N[v_j]$, where $j<i$. It is well known that the Grundy domination number of any graph $G$ is greater than or equal to the upper domination number $Γ(G)$, which is in turn greater than or equal to the independence number $α(G)$. In this paper, we initiate the study of the class of graphs $G$ with $Γ(G)={γ_{\rm gr…
Bicyclic graph is a connected graph in which the number of edges equals the number of vertices plus one. In this paper, we determine the graph which alone maximizes the spectral radii among all the bicyclic graphs on $n$ vertices with fixed independence number.
The $k$-independence number of a graph is the maximum size of a set of vertices at pairwise distance greater than $k$. A graph is called $k$-partially walk-regular if the number of closed walks of a given length $l\le k$, rooted at a vertex $v$, only depends on $l$. In particular, a distance-regular graph is also $k$-partially walk-regular for any $k$. In this note, we introduce a new family of polynomials obtained from the spectrum of a graph. These polynomials, together with the interlacing technique, allow us to give tight spectral bounds on the $k$-independence number of a $k$-partially wa…
If for any k the k-th coefficient of a polynomial I(G;x)is equal to the number of stable sets of cardinality k in graph G, then it is called the independence polynomial of G (Gutman and Harary, 1983). A graph G is very well-covered (Favaron, 1982) if it has no isolated vertices, its order equals 2*alpha(G), where alpha(G) is the size of a maximum stable set, and it is well-covered (i.e., all its maximal independent sets are of the same size, Plummer, 1970). For instance, appending a single pendant edge to each vertex of G yields a very well-covered graph, which we denote by G*. Under certain c…
For $k\ge 1$, the $k$-independence number $α_k$ of a graph is the maximum number of vertices that are mutually at distance greater than $k$. The well-known inertia and ratio bounds for the (1-)independence number $α(=α_1)$ of a graph, due to Cvetković and Hoffman, respectively, were generalized recently for every value of $k$. We show that, for graphs with enough regularity, the polynomials involved in such generalizations are closely related and give exact values for $α_k$, showing a new relationship between the inertia and ratio type bounds. Additionally, we investigate the existence and pro…
In this paper, we obtain two spectral upper bounds for the $k$-independence number of a graph which is is the maximum size of a set of vertices at pairwise distance greater than $k$. We construct graphs that attain equality for our first bound and show that our second bound compares favorably to previous bounds on the $k$-independence number.
The regular independence number, introduced by Albertson and Boutin in 1990, is the size of a largest set of independent vertices with the same degree. Lower bounds were proven for this invariant, in terms of the order, for trees and planar graphs. In this article, we generalize and extend these results to find lower bounds for the regular $k$-independence number for trees, forests, planar graphs, $k$-trees and $k$-degenerate graphs.
Let $G$ be the circulant graph $C_n(S)$ with $S\subseteq\{ 1,\ldots,\left \lfloor\frac{n}{2}\right \rfloor\}$. We study the reduced Euler characteristic $\tildeχ$ of the independence complex $Δ(G)$ for $n=p^k$ with $p$ prime and for $n=2p^k$ with $p$ odd prime, proving that in both cases $\tildeχ$ does not vanish. We also give an example of circulant graph whose independence complex has $\tildeχ$ equals to $0$, giving a negative answer to R. Hoshino.
Recently, Davies, Jenssen, Perkins, and Roberts gave a very nice proof of the result (due, in various parts, to Kahn, Galvin-Tetali, and Zhao) that the independence polynomial of a $d$-regular graph is maximized by disjoint copies of $K_{d,d}$. Their proof uses linear programming bounds on the distribution of a cleverly chosen random variable. In this paper, we use this method to give lower bounds on the independence polynomial of regular graphs. We also give new bounds on the number of independent sets in triangle-free regular graphs.
We leverage an algorithm of Deming [R.W. Deming, Independence numbers of graphs -- an extension of the Koenig-Egervary theorem, Discrete Math., 27(1979), no. 1, 23--33; MR534950] to decompose a matchable graph into subgraphs with a precise structure: they are either spanning even subdivisions of blossom pairs, spanning even subdivisions of the complete graph $K_4$, or a Kőnig-Egerváry graph. In each case, the subgraphs have perfect matchings; in the first two cases, their independence numbers are one less than their matching numbers, while the independence number of the KE subgraph equals its …
An independent set in a graph is a set of pairwise non-adjacent vertices. The independence number $α{(G)}$ is the size of a maximum independent set in the graph $G$. The independence polynomial of a graph is the generating function for the sequence of numbers of independent sets of each size. In other words, the $k$-th coefficient of the independence polynomial equals the number of independent sets comprised of $k$ vertices. For instance, the degree of the independence polynomial of the graph $G$ is equal to $α{(G)}$. In 1987, Alavi, Malde, Schwenk, and Erd{ö}s conjectured that the independenc…
For a digraph $G$ and $v \in V(G)$, let $δ^+(v)$ be the number of out-neighbors of $v$ in $G$. The Caccetta-Häggkvist conjecture states that for all $k \ge 1$, if $G$ is a digraph with $n = |V(G)|$ such that $δ^+(v) \ge n/k$ for all $v \in V(G)$, then G contains a directed cycle of length at most $k$. In [2], N. Lichiardopol proved that this conjecture is true for digraphs with independence number equal to two. In this paper, we generalize that result, proving that the conjecture is true for digraphs with independence number at most $(k+1)/2$.
If for any $k$ the $k$-th coefficient of a polynomial I(G;x) is equal to the number of stable sets of cardinality $k$ in graph $G$, then it is called the independence polynomial of $G$ (Gutman and Harary, 1983). J. I. Brown, K. Dilcher and R. J. Nowakowski (2000) conjectured that the independence polynomial of a well-covered graph $G$ (i.e., a graph whose all maximal independent sets are of the same size) is unimodal, that is, there exists an index $k$ such that the part of the sequence of coefficients from the first to $k$-th is non-decreasing while the other part of coefficients is non-incre…
A distance graph is an undirected graph on the integers where two integers are adjacent if their difference is in a prescribed distance set. The independence ratio of a distance graph $G$ is the maximum density of an independent set in $G$. Lih, Liu, and Zhu [Star extremal circulant graphs, SIAM J. Discrete Math. 12 (1999) 491--499] showed that the independence ratio is equal to the inverse of the fractional chromatic number, thus relating the concept to the well studied question of finding the chromatic number of distance graphs. We prove that the independence ratio of a distance graph is ach…
Studying independent sets of maximum size is equivalent to considering the hard-core model with the fugacity parameter $λ$ tending to infinity. Finding the independence ratio of random $d$-regular graphs for some fixed degree $d$ has received much attention both in random graph theory and in statistical physics. For $d \geq 20$ the problem is conjectured to exhibit 1-step replica symmetry breaking (1-RSB). The corresponding 1-RSB formula for the independence ratio was confirmed for (very) large $d$ in a breakthrough paper by Ding, Sly, and Sun. Furthermore, the so-called interpolation method s…
Fendley, Schoutens and van Eerten [Fendley et al., J. Phys. A: Math. Gen., 38 (2005), pp. 315-322] studied the hard square model at negative activity. They found analytical and numerical evidence that the eigenvalues of the transfer matrix with periodic boundary were all roots of unity. They also conjectured that for an m times n square grid, with doubly periodic boundary, the partition function is equal to 1 when m and n are relatively prime. These conjectures were later proven by Jonsson [Jonsson, Electronic J. Combin., 13(1) (2006), R67]. There, it was also noted that the cylindrical case s…
A dissociation set in a graph is a set of vertices inducing a subgraph of maximum degree at most $1$. Computing the dissociation number ${\rm diss}(G)$ of a given graph $G$, defined as the order of a maximum dissociation set in $G$, is algorithmically hard even when $G$ is restricted to be bipartite. Recently, Hosseinian and Butenko proposed a simple $\frac{4}{3}$-approximation algorithm for the dissociation number problem in bipartite graphs. Their result relies on the inequality ${\rm diss}(G)\leq\frac{4}{3}α(G-M)$ implicit in their work, where $G$ is a bipartite graph, $M$ is a maximum matc…
We estimate the likely values of the chromatic and independence numbers of the random $r$-uniform $d$-regular hypergraph on $n$ vertices for fixed $r$, large fixed $d$, and $n \rightarrow \infty$.
The independence number of a square matrix $A$, denoted by $α(A)$, is the maximum order of its principal zero submatrices. Let $S_n^{+}$ be the set of $n\times n$ nonnegative symmetric matrices with zero trace. Denote by $J_n$ the $n\times n$ matrix with all entries equal to one. Given any integer $n$, we prove that a linear map $ϕ: S_n^+\rightarrow S_n^+$ satisfies $$α(ϕ(X))= α(X) {\quad\rm for~ all\quad}X\in S_n^+$$ if and only if there is a permutation matrix $P$ such that $$ϕ(X)=H\circ(P^TXP)\quad { \rm for~ all\quad}X\in S_n^+,$$ where $H=ϕ(J_n-I_n)$ with all off-diagonal entries positive…
A broadcast on a nontrivial connected graph G is a function f from the vertices of G to the non-negative integers such that f(v) does not exceed e(v) (the eccentricity of v) for each vertex v. If G is disconnected, we define a broadcast on G as the union of broadcasts on its components. In a search for the best way to generalise the concept of independent sets in graphs to independent broadcasts, there are several ways to look at an independent set X of a graph G. One way is from the point of view of the vertices in X: no two vertices are adjacent -- the usual definition. Another way is from t…
Let $G$ be an edge-colored connected graph. A path of $G$ is called rainbow if its every edge is colored by a distinct color. $G$ is called rainbow connected if there exists a rainbow path between every two vertices of $G$. The minimum number of colors that are needed to make $G$ rainbow connected is called the rainbow connection number of $G$, denoted by $rc(G)$. In this paper, we investigate the relation between the rainbow connection number and the independence number of a graph. We show that if $G$ is a connected graph, then $rc(G)\leq 2α(G)-1$. Two examples $G$ are given to show that the …
We count the number and patterns of pairs and tuples of independent events in a simple random experiment: first a fair coin is flipped and then a fair die is tossed. The first number, equal to 888,888, suggest that there are some open questions about the structure of independence even in a finite sample space. We discuss briefly these questions and possible approaches to answer them.
Let $G$ be a graph and $v$ any vertex of $G$. We define the degenerate degree of $v$, denoted by $ζ(v)$ as $ζ(v)={\max}_{H: v\in H}~δ(H)$, where the maximum is taken over all subgraphs of $G$ containing the vertex $v$. We show that the degenerate degree sequence of any graph can be determined by an efficient algorithm. A $k$-independent set in $G$ is any set $S$ of vertices such that $Δ(G[S])\leq k$. The largest cardinality of any $k$-independent set is denoted by $α_k(G)$. For $k\in \{1, 2, 3\}$, we prove that $α_{k-1}(G)\geq {\sum}_{v\in G} \min \{1, 1/(ζ(v)+(1/k))\}$. Using the concept of c…
The residue of a graph is the number of zeros left after iteratively applying the Havel-Hakimi algorithm to its degree sequence. Favaron, Mahéo, and Saclé showed that the residue is a lower bound on the independence number. The Maxine heuristic reduces a graph to an independent set of size $M$. It has been shown that given a graph $G$, $M$ is bounded between the independence number and the residue of a graph for any application of the Maxine heuristic. We improve upon a forbidden subgraph classification of graphs such that $M$ is equal to the independence number given by Barrus and Molnar in 2…
The {\em independence ratio} of a graph $G$ is defined by \[ ι(G) := \sup_{X \subset V(G)} \frac{|X|}{α(X)},\] where $α(X)$ is the independence number of the subgraph of $G$ induced by $X$. The independence ratio is a relaxation of the chromatic number $χ(G)$ in the sense that $χ(G) \geq ι(G)$ for every graph $G$, while for many natural classes of graphs these quantities are almost equal. In this paper, we address two old conjectures of Erdős on cycles in graphs with large chromatic number and a conjecture of Erdős and Hajnal on graphs with infinite chromatic number.
This article considers a strongly regular graph with parameters (2048,276,44,36) that is related to the extended binary Golay code. That graph is known since about 1975 but there seems to be not much information on the contained maximal cocliques. The source package of this article contains a data file that encodes a sequence of maximal independence sets of that graph, covering all sizes from 20 to 67 and the size 72, and a Pascal program to check this assertion and to optionally generate a text file (to be read by the computer algebra system GAP) that contains the adjacency lists of that grap…
Let $i(r,g)$ denote the infimum of the ratio $\frac{α(G)}{|V(G)|}$ over the $r$-regular graphs of girth at least $g$, where $α(G)$ 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 $i(3, \infty) \ge 0.4375$, and Csóka improved it to $i(3,\infty) \ge 0.44533$ 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.…
We propose a new coloring game on a graph, called the independence coloring game, which is played by two players with opposite goals. The result of the game is a proper coloring of vertices of a graph $G$, and Alice's goal is that as few colors as possible are used during the game, while Bob wants to maximize the number of colors. The game consists of rounds, and in round $i$, where $i=1,2,,\ldots$, the players are taking turns in selecting a previously unselected vertex of $G$ and giving it color $i$ (hence, in each round the selected vertices form an independent set). The game ends when all …
A graph $G$ is well-covered if all its maximal stable sets have the same size, denoted by alpha(G) (M. D. Plummer, 1970). If for any $k$ the $k$-th coefficient of a polynomial I(G;x) is equal to the number of stable sets of cardinality $k$ in graph $G$, then it is called the independence polynomial of $G$ (Gutman and Harary, 1983). J. I. Brown, K. Dilcher and R. J. Nowakowski (2000) conjectured that I(G;x) is unimodal (that is, there exists an index $k$ such that the part of the sequence of coefficients from the first to $k$-th is non-decreasing while the other part of coefficients is non-incr…
Let $G$ be a group. The power graph of $G$ is a graph with vertex set $G$ in which two distinct elements $x,y$ are adjacent if one of them is a power of the other. We characterize all groups whose power graphs have finite independence number, show that they have clique cover number equal to their independence number, and calculate this number. The proper power graph is the induced subgraph of the power graph on the set $G-\{1\}$. A group whose proper power graph is connected must be either a torsion group or a torsion-free group; we give characterizations of some groups whose proper power grap…
For a set $S$ of vertices of a graph $G$, a vertex $u$ in $V(G)\setminus S$, and a vertex $v$ in $S$, let ${\rm dist}_{(G,S)}(u,v)$ be the distance of $u$ and $v$ in the graph $G-(S\setminus \{ v\})$. Dankelmann et al. (Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883) define $S$ to be an exponential dominating set of $G$ if $w_{(G,S)}(u)\geq 1$ for every vertex $u$ in $V(G)\setminus S$, where $w_{(G,S)}(u)=\sum\limits_{v\in S}\left(\frac{1}{2}\right)^{{\rm dist}_{(G,S)}(u,v)-1}$. Inspired by this notion, we define $S$ to be an exponential independent set of $G$ if $w_{(G…
We characterize the graphs for which the independence number equals the packing number. As a consequence we obtain simple structural descriptions of the graphs for which (i) the distance-$k$-packing number equals the distance-$2k$-packing number, and (ii) the distance-$k$-matching number equals the distance-$2k$-matching number. This last result considerably simplifies and extends previous results of Cameron and Walker (The graphs with maximum induced matching and maximum matching the same size, Discrete Math. 299 (2005) 49-55). For positive integers $k_1$ and $k_2$ with $k_1<k_2$ and $\lceil(…
Delete the edges of a Kneser graph independently of each other with some probability: for what probabilities is the independence number of this random graph equal to the independence number of the Kneser graph itself? We prove a sharp threshold result for this question in certain regimes. Since an independent set in the Kneser graph is the same as a uniform intersecting family, this gives us a random analogue of the Erdős-Ko-Rado theorem.
A broadcast on a nontrivial connected graph G with vertex set V is a function f from V to {0,1,...,diam(G)} such that f(v) is at most the eccentricity of v for all vertices v. The weight of f is the sum of the function values taken over V. A vertex u hears f from v if f(v) is positive and d(u,v) is at most f(v). A broadcast f is boundary independent if, for any vertex w that hears f from vertices v_{1},...,v_{k}, where k is at least 2, d(w,v_{i}) equals f(v_{i}) for each i. The maximum weight of a boundary independent broadcast on G is denoted by α_{bn}(G). We prove a sharp upper bound on α_{b…
If alpha=alpha(G) is the maximum size of an independent set and s_{k} equals the number of stable sets of cardinality k in graph G, then I(G;x)=s_{0}+s_{1}x+...+s_{alpha}x^{alpha} is the independence polynomial of G. In this paper we prove that: 1. I(T;-1) equels either -1 or 0 or 1 for every tree T; 2. I(G;-1)=0 for every connected well-covered graph G of girth > 5, non-isomorphic to C_{7} or K_{2}; 3. the absolute value of I(G;-1) is not greater than 2^nu(G), for every graph G, where nu(G) is its cyclomatic number.
A theorem of Hoffman gives an upper bound on the independence ratio of regular graphs in terms of the minimum $λ_{\min}$ of the spectrum of the adjacency matrix. To complement this result we use random eigenvectors to gain lower bounds in the vertex-transitive case. For example, we prove that the independence ratio of a $3$-regular transitive graph is at least \[q=\frac{1}{2}-\frac{3}{4π}\arccos\biggl(\frac{1-λ_{\min}}{4}\biggr).\] The same bound holds for infinite transitive graphs: we construct factor of i.i.d. independent sets for which the probability that any given vertex is in the set is…
We show that there are polynomial-time algorithms to compute maximum independent sets in the categorical products of two cographs and two splitgraphs. The ultimate categorical independence ratio of a graph G is defined as lim_{k --> infty} α(G^k)/n^k. The ultimate categorical independence ratio is polynomial for cographs, permutation graphs, interval graphs, graphs of bounded treewidth and splitgraphs. When G is a planar graph of maximal degree three then alpha(G \times K_4) is NP-complete. We present a PTAS for the ultimate categorical independence ratio of planar graphs. We present an O^*(n^…
