Related papers: Disconnected Cliques in Derangement Graphs
Chordal decomposition techniques are used to reduce large structured positive semidefinite matrix constraints in semidefinite programs (SDPs). The resulting equivalent problem contains multiple smaller constraints on the nonzero blocks (or…
A clique in an undirected graph G= (V, E) is a subset V' V of vertices, each pair of which is connected by an edge in E. The clique problem is an optimization problem of finding a clique of maximum size in graph. The clique problem is…
It is well-known that the graph isomorphism problem can be posed as an equivalent problem of determining whether an auxiliary graph structure contains a clique of specific order. However, the algorithms that have been developed so far for…
In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph…
In extremal graph theory, the problem of finding the elements of a given class of graphs which contain the most cliques traces its routes back to Tur\'an's famous theorem. We consider the implications of the connectivity property of…
We study the problem of finding pairwise vertex-disjoint triangles in the randomly perturbed graph model, which is the union of any $n$-vertex graph $G$ satisfying a given minimum degree condition and the binomial random graph $G(n,p)$. We…
We describe two constructions of (very) dense graphs which are edge disjoint unions of large {\em induced} matchings. The first construction exhibits graphs on $N$ vertices with ${N \choose 2}-o(N^2)$ edges, which can be decomposed into…
For a positive integer $s$, an $s$-club in a graph $G$ is a set of vertices inducing a subgraph with diameter at most $s$. As generalizations of cliques, $s$-clubs offer a flexible model for real-world networks. This paper addresses the…
We study the problem of finding a maximum-cardinality set of $r$-cliques in an undirected graph of fixed maximum degree $\Delta$, subject to the cliques in that set being either vertex-disjoint or edge-disjoint. It is known for $r=3$ that…
The maximum clique problem is a classical NP-complete problem in graph theory and has important applications in many domains. In this paper we show, in a partially non-constructive way, the existence of an exact polynomial-time algorithm…
The advent of new special-purpose hardware such as FPGA or ASIC-based annealers and quantum processors has shown potential in solving certain families of complex combinatorial optimization problems more efficiently than conventional CPUs.…
The maximal clique problem, to find the maximally sized clique in a given graph, is classically an NP-complete computational problem, which has potential applications ranging from electrical engineering, computational chemistry,…
In this paper, we relate the problem of finding a maximum clique to the intersection number of the input graph (i.e. the minimum number of cliques needed to edge cover the graph). In particular, we consider the maximum clique problem for…
We study graphs whose chromatic number is close to the order of the graph (the number of vertices). Both when the chromatic number is a constant multiple of the order and when the difference of the chromatic number and the order is a small…
Generalized Tur\'an problems have been a central topic of study in extremal combinatorics throughout the last few decades. One such problem, maximizing the number of cliques of a fixed order in a graph with fixed number of vertices and…
A geometric intersection graph is constructed over a set of geometric objects, where each vertex represents a distinct object and an edge connects two vertices if and only if the corresponding objects intersect. We examine the problem of…
Finding maximum cliques in large networks is a challenging combinatorial problem with many real-world applications. We present a fast algorithm to achieve the exact solution for the maximum clique problem in large sparse networks based on…
The (Perfect) Matching Cut problem is to decide if a connected graph has a (perfect) matching that is also an edge cut. The Disconnected Perfect Matching problem is to decide if a connected graph has a perfect matching that contains a…
The theory of dense graph limits comes with a natural sampling process which yields an inhomogeneous variant G(n,W) of the Erdos-Renyi random graph. Here we study the clique number of these random graphs. We establish the concentration of…
We prove that any $n$-vertex graph whose complement is triangle-free contains $n^2/12-o(n^2)$ edge-disjoint triangles. This is tight for the disjoint union of two cliques of order $n/2$. We also prove a corresponding stability theorem, that…