Gallai's path decomposition conjecture states that if $G$ is a connected graph on $n$ vertices, then the edges of can be decomposed into at most $\lceil \frac{n }{2} \rceil$ paths. A said to an odd semi-clique it obtained from clique $2k+1$ vertices by deleting $k-1$ edges. Bonamy and Perrett asked every $\lfloor \frac{n}{2} \rfloor$ paths unless semi-clique. 2-degenerate subgraph has vertex de...

‎in this paper‎, ‎we give some necessary conditions for an $r$-partite graph such that the edge ring of the graph is cohen-macaulay‎. ‎it is proved that if there exists a cover of an $r$-partite cohen-macaulay graph by disjoint cliques of size $r$‎, ‎then such a cover is unique‎.

In this paper, we study the problem of designing in-place algorithms for finding the maximum clique in the intersection graphs of axis-parallel rectangles and disks in R2. First, we propose an O(n2 log n) time in-place algorithm for finding the maximum clique of the intersection graph of a set of n axis-parallel rectangles of arbitrary sizes. For the intersection graph of fixed height rectangle...

In 1994 S. McGuinness showed that any greedy clique decomposition of an n-vertex graph has at most ⌊n2/4⌋ cliques (The greedy clique decomposition of a graph, J. Graph Theory 18 (1994) 427-430), where a clique decomposition means a clique partition of the edge set and a greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empt...

Clique-width is a graph parameter, defined by a composition mechanism for vertexlabeled graphs, which measures in a certain sense the complexity of a graph. Hard graph problems (e.g., problems expressible in Monadic Second Order Logic, that includes NPhard problems) can be solved efficiently for graphs of certified small clique-width. It is widely believed that determining the clique-width of a...

We propose the n-clique network as a powerful tool for understanding global structures of combined highly-interconnected subgraphs, and provide theoretical predictions for statistical properties of the n-clique networks embedded in a complex network using the degree distribution and the clustering spectrum. Furthermore, using our theoretical predictions, we find that the statistical properties ...

A clique is a subgraph in which any two vertices are connected. Clique represents a densely connected structure in the graph, thus used to capture the local related elements such as clustering, frequent patterns, community mining, and so on. In these applications, enumeration of cliques rather than optimization is frequently used. Recent applications have large scale very sparse graphs, thus ef...

We characterize clique trees of a chordal graph in their relation to simplicial vertices and perfect sequences of maximal cliques. We investigate boundary cliques de ned by Shibata[23] and clarify their relation to endpoints of clique trees. Next we de ne a symmetric binary relation between the set of clique trees and the set of perfect sequences of maximal cliques. We describe the relation as ...

The k-fold clique transversal problem is to locate a minimum set of vertices of a graph such that every maximal clique has at least k elements of. The maximum h-colourable subgraph problem is to nd a maximum subgraph of a graph which is h-colourable. We show that the k-fold clique transversal problem and the maximum h-colourable subgraph problem are polynomially solvable on balanced graphs. We ...

Does well-quasi-ordering by induced subgraphs imply bounded clique-width for hereditary classes? This question was asked by Daligault, Rao and Thomassé in [7]. We answer this question negatively by presenting a hereditary class of graphs of unbounded clique-width which is well-quasi-ordered by the induced subgraph relation. We also show that graphs in our class have at most logarithmic clique-w...