A graph $G$ is $H$-covered by some given graph $H$ if each vertex in $G$ is contained in a copy of $H$. In this note, we give the maximum number of independent sets of size $t\ge 3$ in $K_n$-covered graphs of size $N\ge n+t-1$ and determine its extremal graph. The result answers a question proposed by Chakraborit and Loh. The proof uses an edge-switching operation of hypergraphs which remains the number of independent sets nondecreasing.
There has been interest recently in maximizing the number of independent sets in graphs. For example, the Kahn-Zhao theorem gives an upper bound on the number of independent sets in a $d$-regular graph. Similarly, it is a corollary of the Kruskal-Katona theorem that the lex graph has the maximum number of independent sets in a graph of fixed size and order. In this paper we solve two equivalent problems. The first is: what $3$-uniform hypergraph on a ground set of size $n$, having at least $t$ edges, has the most $2$-independent sets? Here a $2$--independent set is a subset of vertices contain…
For a fixed graph G, a maximal independent set is an independent set that is not a proper subset of any other independent set. P. Erdös, and independently, J. W. Moon and L. Moser, and R. E. Miller and D. E. Muller, determined the maximum number of maximal independent sets in a graph on n vertices, as well as the extremal graphs. In this paper we maximize the number of maximal independent sets of a fixed size for all graphs of order n and determine the extremal graphs. Our result generalizes the classical result.
Let $i_t(G)$ be the number of independent sets of size $t$ in a graph $G$. Engbers and Galvin asked how large $i_t(G)$ could be in graphs with minimum degree at least $δ$. They further conjectured that when $n\geq 2δ$ and $t\geq 3$, $i_t(G)$ is maximized by the complete bipartite graph $K_{δ, n-δ}$. This conjecture has drawn the attention of many researchers recently. In this short note, we prove this conjecture.
Let $G=(V,E)$ be a matching-covered graph and $X$ be an edge set of $G$. $X$ is said to be feasible if there exist two perfect matchings $M_1$ and $M_2$ in $G$ such that $|M_1\cap X|\not \equiv|M_2\cap X|\ (\mbox{mod } 2)$. For any $V_0\subseteq V$, $X$ is said to be switching-equivalent to $X\oplus \nabla_G(V_0)$, where $\nabla_G(V_0)$ is the set of edges in $G$ each of which has exactly one end in $V_0$ and $A \oplus B$ is the symmetric difference of two sets $A$ and $B$. Lukot'ka and Rollová showed that when $G$ is regular and bipartite, $X$ is non-feasible if and only if $X$ is switching-e…
The main purpose of this paper is to prove the uniqueness of a graph attaining the maximum of the number of independent sets over all $k$-regular graphs on $n$ vertices for $2k|n$.
A well known consequence of the Borsuk-Ulam theorem is that if the $d$-dimensional sphere $S^d$ is covered with less than $d+2$ open sets, then there is a set containing a pair of antipodal points. In this paper we provide lower and upper bounds on the minimum number of open sets, not containing a pair of antipodal points, needed to cover the $d$-dimensional sphere $n$ times, with the additional property that the northern hemisphere is covered $m > n$ times. We prove that if the open northern hemisphere is to be covered $m$ times then at least $ \lceil \frac{d-1}{2} \rceil+n+m$ and at most $d+…
We study the problem of maximizing the number of independent sets in $n$-vertex $k$-chromatic $\ell$-connected graphs. First we consider maximizing the total number of independent sets in such graphs with $n$ sufficiently large, and for this problem we use a stability argument to find the unique extremal graph. We show that our result holds within the larger family of $n$-vertex $k$-chromatic graphs with minimum degree at least $\ell$, again for $n$ sufficiently large. We also maximize the number of independent sets of each fixed size in $n$-vertex 3-chromatic 2-connected graphs. We finally ad…
A maximum stable set in a graph G is a stable set of maximum cardinality. S is called a local maximum stable set of G if S is a maximum stable set of the subgraph induced by the closed neighborhood of S. A greedoid (V,F) is called a local maximum stable set greedoid if there exists a graph G=(V,E) such that its family of local maximum stable sets coinsides with (V,F). It has been shown that the family local maximum stable sets of a forest T forms a greedoid on its vertex set. In this paper we demonstrate that if G is a very well-covered graph, then its family of local maximum stable sets is a …
A graph is well-covered if all its maximal independent sets are of the same size (M. D. Plummer, 1970). A well-covered graph is 1-well-covered if the deletion of every vertex leaves a graph which is well-covered as well (J. W. Staples, 1975). A graph G belongs to class W_{n} if every n pairwise disjoint independent sets in G are included in $n$ pairwise disjoint maximum independent sets (J. W. Staples, 1975). Clearly, W_{1} is the family of all well-covered graphs. It turns out that G belongs to W_{2} if and only if it is a 1-well-covered graph without isolated vertices. We show that deleting …
Let F be a finite nonempty family of finite nonempty sets. We prove the following: (i) F satisfies the condition of the title if and only if for every pair of distinct subfamilies {A_1,...,A_r}, {B_1,...,B_s} of F, the union of the A_i is different from the union of the B_i. (ii) If F satisfies the condition of the title, then the number of subsets of the union of the members of F containing at least one set of F is odd. We give two applications of these results, one to number theory and one to commutative algebra.
Given $2k-1$ convex sets in $R^2$ such that no point of the plane is covered by more than $k$ of the sets, is it true that there are two among the convex sets whose union contains all $k$-covered points of the plane? This question due to Gy. Petruska has an obvious affirmative answer for $k=1,2,3$; we show here that the claim is also true for $k=4$, and we present a counterexample for $k=5$. We explain how Petruska's geometry question fits into the classical hypergraph extremal problems, called arrow problems, proposed by P. Erdős.
In 1970, Plummer defined a well-covered graph to be a graph $G$ in which all maximal independent sets are in fact maximum. Later Hartnell and Rall showed that if the Cartesian product $G \Box H$ is well-covered, then at least one of $G$ or $H$ is well-covered. In this paper, we consider the problem of classifying all well-covered Cartesian products. In particular, we show that if the Cartesian product of two nontrivial, connected graphs of girth at least $4$ is well-covered, then at least one of the graphs is $K_2$. Moreover, we show that $K_2 \Box K_2$ and $C_5 \Box K_2$ are the only well-cov…
The independence polynomial $i(G,x)$ of a graph $G$ is the generating function of the numbers of independent sets of each size. A graph of order $n$ is very well-covered if every maximal independent set has size $n/2$. Levit and Mandrescu conjectured that the independence polynomial of every very well-covered graph is unimodal (that is, the sequence of coefficients is nondecreasing, then nonincreasing). In this article we show that every graph is embeddable as an induced subgraph of a very well-covered graph whose independence polynomial is unimodal, by considering the location of the roots of…
The $k$-token graph $T_k(G)$ is the graph whose vertices are the $k$-subsets of vertices of a graph $G$, with two vertices of $T_k(G)$ adjacent if their symmetric difference is an edge of $G$. We explore when $T_k(G)$ is a well-covered graph, that is, when all of its maximal independent sets have the same cardinality. For bipartite graphs $G$, we classify when $T_k(G)$ is well-covered. For an arbitrary graph $G$, we show that if $T_2(G)$ is well-covered, then the girth of $G$ is at most four. We include upper and lower bounds on the independence number of $T_k(G)$, and provide some families of…
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…
The number of stable sets of cardinality $k$ in graph $G$ is the $k$-th coefficient of the independence polynomial of $G$ (I. Gutman and F. Harary, 1983). In 1990, Y. O. Hamidoune proved that for any claw-free graph, its independence polynomial is unimodal, i.e., there exists a coefficient $k$ such that the part of the sequence of coefficients from the first to $k$-th is non-decreasing while the second part of coefficients is non-increasing. Y. Alavi, P. J. Malde, A. J. Schwenk and P. Erdös (1987) asked whether for trees (or perhaps forests) the independence polynomial is unimodal. J. I. Brown…
A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space. Given an input graph G without cycles of length 4, 5, and 6, we characterize polynomially the vector space of weight functions w for which G is w-well-covered. Let B be an induced complete bipartite subgraph of G on vertex sets of bipartition B_{X} and B_{Y}. Assume tha…
A graph $G$ is well-covered if all maximal independent sets of $G$ have the same cardinality. In 1992 Topp and Volkmann investigated the structure of well-covered graphs that have nontrivial factorizations with respect to some of the standard graph products. In particular, they showed that both factors of a well-covered direct product are also well-covered and proved that the direct product of two complete graphs (respectively, two cycles) is well-covered precisely when they have the same order (respectively, both have order 3 or 4). Furthermore, they proved that the direct product of two well…
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…
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…
We determine, within 1, the value of N for which sum (s1 choose i)(s2 choose N)(s1 choose N-i)(N choose i) achieves its maximum value. Here s1 and s2 are fixed integers. This problem arises in studying the most likely value for the size of the union of A, B, and C if A and C are disjoint sets of size s1, and B is a set of size s2. Attempting to remove the 1 unit of indeterminacy leads to interesting conjectures about a family of rational functions.
We show for each positive integer $a$ that, if $\mathcal{M}$ is a minor-closed class of matroids not containing all rank-$(a+1)$ uniform matroids, then there exists an integer $c$ such that either every rank-$r$ matroid in $\mathcal{M}$ can be covered by at most $r^c$ rank-$a$ sets, or $\mathcal{M}$ contains the GF$(q)$-representable matroids for some prime power $q$ and every rank-$r$ matroid in $\mathcal{M}$ can be covered by at most $cq^r$ rank-$a$ sets. In the latter case, this determines the maximum density of matroids in $\mathcal{M}$ up to a constant factor.
We show for each positive integer $a$ that, if $\cM$ is a minor-closed class of matroids not containing all rank-$(a+1)$ uniform matroids, then there exists an integer $n$ such that either every rank-$r$ matroid in $\cM$ can be covered by at most $r^n$ sets of rank at most $a$, or $\cM$ contains the $\GF(q)$-representable matroids for some prime power $q$, and every rank-$r$ matroid in $\cM$ can be covered by at most $r^nq^r$ sets of rank at most $a$. This determines the maximum density of the matroids in $\cM$ up to a polynomial factor.
A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space. Given an input claw-free graph G, we present an O(n^6)algortihm, whose input is a claw-free graph G, and output is the vector space of weight functions w, for which G is w-well-covered. A graph G is equimatchable if all its maximal matchings are of the same cardinality.…
A corollary of Kneser's theorem, one sees that any finite non-empty subset $A$ of an abelian group $G = (G,+)$ with $|A + A| \leq (2-\eps) |A|$ can be covered by at most $\frac{2}{\eps}-1$ translates of a finite group $H$ of cardinality at most $(2-\eps)|A|$. Using some arguments of Hamidoune, we establish an analogue in the noncommutative setting. Namely, if $A$ is a finite non-empty subset of a nonabelian group $G = (G,\cdot)$ such that $|A \cdot A| \leq (2-\eps) |A|$, then $A$ is either contained in a right-coset of a finite group $H$ of cardinality at most $\frac{2}{\eps}|A|$, or can be co…
A graph $G$ is called well-covered if all maximal independent sets of vertices have the same cardinality. A well-covered graph $G$ is called uniformly well-covered if there is a partition of the set of vertices of $G$ such that each maximal independent set of vertices has exactly one vertex in common with each part in the partition. The problem of determining which graphs is well-covered, was proposed in 1970 by M.D. Plummer. Let $\cal G$ be the class of graphs with some disjoint maximal cliques covering all vertices. In this paper, some necessary and sufficient conditions are presented to rec…
A graph is well-covered if all its maximal independent sets are of the same cardinality (Plummer, 1970). If G is a well-covered graph, has at least two vertices, and G-v is well-covered for every vertex v, then G is a 1-well-covered graph (Staples, 1975). We call G a λ-quasi-regularizable graph if λ |S| =< |N(S)| for every independent set S of G. The independence polynomial I(G;x) is the generating function of independent sets in a graph G (Gutman & Harary, 1983). The Roller-Coaster Conjecture (Michael & Travis, 2003), saying that for every permutation σ of the set {(α/2),...,α} there exists a…
A graph $G$ is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function $w$ is defined on its vertices. Then $G$ is $w$-well-covered if all maximal independent sets are of the same weight. For every graph $G$, the set of weight functions $w$ such that $G$ is $w$-well-covered is a vector space, denoted $WCW(G)$. Let $B$ be a complete bipartite induced subgraph of $G$ on vertex sets of bipartition $B_{X}$ and $B_{Y}$. Then $B$ is generating if there exists an independent set $S$ such that $S \cup B_{X}$ and $S \cup B_{Y}$ are both maximal indepe…
A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space, denoted WCW(G). Let B be a complete bipartite induced subgraph of G on vertex sets of bipartition B_X and B_Y. Then B is generating if there exists an independent set S such that S \cup B_X and S \cup B_Y are both maximal independent sets of G. A relating edge is a gene…
A graph $G$ is well-covered if it has no isolated vertices and all the maximal independent sets have the same cardinality. If furthermore two times this cardinality is equal to $|V(G)|$, the graph $G$ is called very well-covered. The class of very well-covered graphs contains bipartite well-covered graphs. Recently in \cite{CRT} it is shown that a very well-covered graph $G$ is Cohen-Macaulay if and only if it is pure shellable. In this article we improve this result by showing that $G$ is Cohen-Macaulay if and only if it is pure vertex decomposable. In addition, if $I(G)$ denotes the edge ide…
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 the graph $G$, then it is called the independence polynomial of $G$ (Gutman and Harary, 1983). Let $a$ be the size of a maximum stable set. Alavi, Malde, Schwenk and Erdos (1987)conjectured that I(T,x) is unimodal for any tree T, while, in general, they proved that for any permutation $p$ of {1,2,...,a} there is a graph such that s_{p(1)}<s_{p(2)}<...<s_{p(a)}. Brown, Dilcher and Nowakowski (2000) conjectured that I(G;x) is unimodal for any well-covered graph. Michael and Tr…
A graph is said to be well-covered if all its maximal independent sets are of the same size. In 1999, Yamashita and Kameda introduced a subclass of well-covered graphs, called localizable graphs and defined as graphs having a partition of the vertex set into strong cliques, where a clique in a graph is strong if it intersects all maximal independent sets. Yamashita and Kameda observed that all well-covered trees are localizable, pointed out that the converse inclusion fails in general, and asked for a characterization of localizable graphs. In this paper we obtain several structural and algori…
Maximizing $K_r + I_r$ in graphs with fixed edge density
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Institute for Basic Science (IBS)（韩国基础科学研究院）；Jagiellonian University（雅盖隆大学）；Iowa State University（爱荷华州立大学）；University of South Florida（南佛罗里达大学）；University of Illinois, Chicago（伊利诺伊大学芝加哥分校）
本研究确定了固定边密度的大图中诱导团或独立集的 $r$ 元子集最大比例的渐近值，推广了 Olpp 的 $r=3$ 结果，并利用 AI 辅助完成证明。
A graph $G$ is {\em well-covered} if every maximal independent set has the same cardinality $q$. Let $i_k(G)$ denote the number of independent sets of cardinality $k$ in $G$. Brown, Dilcher, and Nowakowski conjectured that the independence sequence $(i_0(G), i_1(G), \ldots, i_q(G))$ was unimodal for any well-ordered graph $G$ with independence number $q$. Michael and Traves disproved this conjecture. Instead they posited the so-called ``Roller Coaster" Conjecture: that the terms \[ i_{\left\lceil\frac{q}2\right\rceil}(G), i_{\left\lceil\frac{q}2\right\rceil+1}(G), \ldots, i_q(G) \] could be in…
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 the graph $G$, then it is called the independence polynomial of $G$ (Gutman and Harary, 1983). Alavi, Malde, Schwenk and Erdos (1987) conjectured that $I(G;x)$ is unimodal, whenever $G$ is a forest, while Brown, Dilcher and Nowakowski (2000) conjectured that $I(G;x)$ is unimodal for any well-covered graph G. Michael and Traves (2003) showed that the assertion is false for well-covered graphs with $a(G)$ > 3 ($a(G)$ is the size of a maximum stable set of the graph $G$), whi…
For a matroid $M$ having $m$ rank-one flats, the density $d(M)$ is $\tfrac{m}{r(M)}$ unless $m = 0$, in which case $d(M)= 0$. A matroid is density-critical if all of its proper minors of non-zero rank have lower density. By a 1965 theorem of Edmonds, a matroid that is minor-minimal among simple matroids that cannot be covered by $k$ independent sets is density-critical. It is straightforward to show that $U_{1,k+1}$ is the only minor-minimal loopless matroid with no covering by $k$ independent sets. We prove that there are exactly ten minor-minimal simple obstructions to a matroid being able t…
A CIS graph is a graph in which every maximal stable set and every maximal clique intersect. A graph is well-covered if all its maximal stable sets are of the same size, co-well-covered if its complement is well-covered, and vertex-transitive if, for every pair of vertices, there exists an automorphism of the graph mapping one to the other. We show that a vertex-transitive graph is CIS if and only if it is well-covered, co-well-covered, and the product of its clique and stability numbers equals its order. A graph is irreducible if no two distinct vertices have the same neighborhood. We classif…
Many extremal problems for graphs have threshold graphs as their extremal examples. For instance the current authors proved that for fixed $k\ge 1$, among all graphs on $n$ vertices with $m$ edges, some threshold graph has the fewest matchings of size $k$; indeed either the lex graph or the colex graph is such an extremal example. In this paper we consider the problem of maximizing the number of matchings in the class of threshold graphs. We prove that the minimizers are what we call \emph{almost alternating threshold graphs}. We also discuss a problem with a similar flavor: which threshold gr…
We prove a tight upper bound on the independence polynomial (and total number of independent sets) of cubic graphs of girth at least 5. The bound is achieved by unions of the Heawood graph, the point/line incidence graph of the Fano plane. We also give a tight lower bound on the total number of independent sets of triangle-free cubic graphs. This bound is achieved by unions of the Petersen graph. We conjecture that in fact all Moore graphs are extremal for the scaled number of independent sets in regular graphs of a given minimum girth, maximizing this quantity if their girth is even and minim…
A graph $G$ is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function $w$ is defined on its vertices. Then $G$ is $w$-well-covered if all maximal independent sets are of the same weight. For every graph $G$, the set of weight functions $w$ such that $G$ is $w$-well-covered is a vector space, denoted as $WCW(G).$ Deciding whether an input graph $G$ is well-covered is co-NP-complete. Therefore, finding $WCW(G)$ is co-NP-hard. A generating subgraph of a graph $G$ is an induced complete bipartite subgraph $B$ of $G$ on vertex sets of bipartition…
A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set of G, if S is a maximum stable set of the subgraph induced by its closed neighborhood. It is known that the family of all local maximum stable sets of a forest forms a greedoid on its vertex set. Bipartite, triangle-free, and well-covered graphs whose families of local maximum stable sets form greedoids have been analyzed as well. A unicycle graph owns only one cycle. In this paper we characterize the unicycle graphs whose families of local maximum stable sets form greedoids.
Let $B$ be an induced complete bipartite subgraph of $G$ on vertex sets of bipartition $B_{X}$ and $B_{Y}$. The subgraph $B$ is {\it generating} if there exists an independent set $S$ such that each of $S \cup B_{X}$ and $S \cup B_{Y}$ is a maximal independent set in the graph. If $B$ is generating, it \textit{produces} the restriction $w(B_{X})=w(B_{Y})$. Let $w:V(G) \longrightarrow\mathbb{R}$ be a weight function. We say that $G$ is $w$-well-covered if all maximal independent sets are of the same weight. The graph $G$ is $w$-well-covered if and only if $w$ satisfies all restrictions produced…
Motivated by an old conjecture of P. Erdős and V. Neumann-Lara, our aim is to investigate digraphs with uncountable dichromatic number and orientations of undirected graphs with uncountable chromatic number. A graph has uncountable chromatic number if its vertices cannot be covered by countably many independent sets, and a digraph has uncountable dichromatic number if its vertices cannot be covered by countably many acyclic sets. We prove that consistently there are digraphs with uncountable dichromatic number and arbitrarily large digirth; this is in surprising contrast with the undirected ca…
A maximum stable set in a graph G is a stable set of maximum cardinality. S is a local maximum stable set of G, if S is a maximum stable set of the subgraph induced by its closed neighborhood. Nemhauser and Trotter Jr. proved in 1975 that any local maximum stable set is a subset of a maximum stable set of G. In 2002 we showed that the family of all local maximum stable sets of a forest forms a greedoid on its vertex set. The cases where G is bipartite, triangle-free, well-covered, while the family of all local maximum stable sets is a greedoid, were analyzed in 2004, 2007, and 2008, respective…
