Related papers: A conjecture on independent sets and graph covers
Independent set games are cooperative games defined on graphs, where players are edges and the value of a coalition is the maximum cardinality of independent sets in the subgraph defined by the coalition. In this paper, we investigate the…
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…
Frankl's union-closed sets conjecture states that in every finite union-closed set of sets, there is an element that is contained in at least half of the member-sets (provided there are at least two members). The conjecture has an…
We enumerate the independent sets of several classes of regular and almost regular graphs and compute the corresponding generating functions. We also note the relations between these graphs and other combinatorial objects and, in some…
The number of independent sets in regular bipartite expander graphs can be efficiently approximated by expressing it as the partition function of a suitable polymer model and truncating its cluster expansion. While this approach has been…
An independent set A is maximal if it is not a proper subset of an independent set, while A is maximum if it has a maximum size. The problem of whether a graph has a pair of disjoint maximal independent sets was introduced by C. Berge in…
We provide a simple proof for the union-closed sets conjecture, a long-standing open problem in set theory with immediate applications to graph theory, number theory, and order-theory.
Probabilistic independence can dramatically simplify the task of eliciting, representing, and computing with probabilities in large domains. A key technique in achieving these benefits is the idea of graphical modeling. We survey existing…
Beck's conjecture on coloring of graphs associated to various algebraic objects has generated considerable interest in the community of discrete mathematics and combinatorics since its inception in the year 1988. The version of this…
In this paper we provide an asymptotic expansion for the number of independent sets in a general class of regular, bipartite graphs satisfying some vertex-expansion properties, extending results of Jenssen and Perkins on the hypercube and…
We develop a new method for enumerating independent sets of a fixed size in general graphs, and we use this method to show that a conjecture of Engbers and Galvin holds for all but finitely many graphs. We also use our method to prove…
We introduce a new graph polynomial that encodes interesting properties of graphs, for example, the number of matchings and the number of perfect matchings. Most importantly, for bipartite graphs the polynomial encodes the number of…
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.…
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…
The independent domination number of a finite graph G is the minimum cardinality of an independent dominating set of vertices. The independent bondage number of G is the minimum cardinality of a set of edges whose deletion results in a…
A conjecture regarding the structure of expander graphs is discussed.
We present two related conjectures, arising in work on i-matchings in random r-regular bipartite graphs. The conjectures themselves are easily stated and involve only basic properties of convergent power series. One formulation involves…
An independent dominating set of a graph, also known as a maximal independent set, is a set $S$ of pairwise non-adjacent vertices such that every vertex not in $S$ is adjacent to some vertex in $S$. We prove that for $\Delta=4$ or…
We prove the automorphism conjecture for ordered sets of width less than or equal to 11. The proof supports the meta conjecture that a large number of automorphisms is achievable only as some type of product of independent automorphisms on…
The notion of graph cover, also known as locally bijective homomorphism, is a discretization of covering spaces known from general topology. It is a pair of incidence-preserving vertex- and edge-mappings between two graphs, the…