We study reconfiguration problems for cliques in a graph, which determine\nwhether there exists a sequence of cliques that transforms a given clique into\nanother one in a step-by-step fashion. As one step of a transformation, we\nconsider three different types of rules, which are defined ...
A clique in a graph is a set of pairwise adjacent vertices. A subset UV is obedient if U is the union of a clique of B and a clique of R. Our first result is that if B has no induced cycles of length four, and R has no induced cycles of length four or five, then every ...
As in (1), denote byKra complete graph withrpoints and denote bykr(H) the number ofKr's in a graphH. A maximal complete subgraph is called aclique.In (1) one of us estimated theminimumofkr(H) providedHhasnpoints andmedges. In this note we shall look at the random variables the...
INTERSECTION graph theoryCHARTS, diagrams, etc.RANDOM graphsGEOMETRYMany real-world networks were found to be highly clustered and contain a large amount of small cliques. We here investigate the number of cliques of any size |$k$| contained in a geometric inhomogeneous random graph: a sca...
For a graph G and a fixed integer k ≥ 3, let v_k(G) denote the maximum number of pairwise edge-disjoint copies of K_k in G. For a constant c, let η(k, c) be the infimum over all constants γ such that any graph G of order n and minimum degree at least cn has v_k(...
Finally, we obtain the extremalnumber of s-cliques in a graph with no path on k-vertices.Mathematics Subject Classif i cation: 05C35, 05C38.Keywords: Turán problem, cycles, paths.1 IntroductionIn [4], Erd˝ os and Gallai determined ex(n,P k ), the maximum number of edges in an ...
Abstract A chordal graph has a dominating clique iff it has diameter at most 3. A strongly chordal graph which has a dominating clique has one as small as the smallest dominating set—and, furthermore, there is a linear-time algorithm to find such a small dominating clique....
Summary: We prove a decomposition theorem for the class $\\cal G$ of $\\{P_{2} \\cup P_{3},C_{4}\\}$-free graphs. This theorem enables us to show that (i) every graph $G$ in $\\cal G$ has at most $n+5$ maximal cliques where $n$ is the number of vertices in ...
N Alon,A Kostochka,C Shikhelman 摘要: For two fixed graphs $T$ and $H$ let $ex(G(n,p),T,H)$ be the random variable counting the maximum number of copies of $T$ in an $H$-free subgraph of the random graph $G(n,p)$. We show that for the case $T=K_m$ and $\\chi(...
On the partition and coloring of a graph by cliques - Wallis, Zhang - 1993 () Citation Context ...in G is a mapping from P to C, such that cliques sharing a vertex have different colors. Let the clique coloring index c(G) be the smallest h such that there is a partition P ...