Related papers: Packing independent cliques into planar graphs

Cliques are defined as complete graphs or subgraphs; they are the strongest form of cohesive subgroup, and are of interest in both social science and engineering contexts. In this paper we show how to efficiently estimate the distribution…

For fixed $k\ge 2$, determining the order of magnitude of the number of edges in an $n$-vertex bipartite graph not containing $C_{2k}$, the cycle of length $2k$, is a long-standing open problem. We consider an extension of this problem to…

For $r \ge 2$ and a graph $G$, let $\alpha_{{r}}(G)$ be the maximum number of vertices in a $K_r$-free subgraph of $G$. We investigate the value $\alpha_{r}(G)$ when $G$ is the random graph $G \sim G_{n, 1/2}$ and discover the following…

In this paper we calculate the average number of cliques in random scale-free networks. We consider first the hidden variable ensemble and subsequently the Molloy Reed ensemble. In both cases we find that cliques, i.e. fully connected…

A $k$-block in a graph $G$ is a maximal set of at least $k$ vertices no two of which can be separated in $G$ by deleting fewer than $k$ vertices. The block number $\beta(G)$ of $G$ is the maximum integer $k$ for which $G$ contains a…

We describe the structure of those graphs that have largest spectral radius in the class of all connected graphs with a given degree sequence. We show that in such a graph the degree sequence is non-increasing with respect to an ordering of…

In the game of pegging, each vertex of a graph is considered a hole into which a peg can be placed. A pegging move is performed by jumping one peg over another peg, and then removing the peg that has been jumped over from the graph. We…

The ring of graph invariants is spanned by the basic graph invariants which calculate the number of subgraphs isomorphic to a given graph in other graphs. These subgraphs counting invariants are not algebraically independent. In our view…

Extremal problems involving independent sets are much studied. Two of the most important extremal problems in this context are concerned with the sharp upper bounds for the number of independent sets of fixed size and the independence…

We find precise asymptotic estimates for the number of planar maps and graphs with a condition on the minimum degree, and properties of random graphs from these classes. In particular we show that the size of the largest tree attached to…

Let $G$ be a non-abelian group. The non-commuting graph $\mathcal{A}_G$ of $G$ is defined as the graph whose vertex set is the non-central elements of $G$ and two vertices are joint if and only if they do not commute. In a finite simple…

The spread of a graph is the difference between the largest and most negative eigenvalue of its adjacency matrix. We show that for sufficiently large $n$, the $n$-vertex outerplanar graph with maximum spread is a vertex joined to a linear…

Extremal problems related to the enumeration of graph substructures, such as independent sets, matchings, and induced matchings, have become a prominent area of research with the advancement of graph theory. A subset of vertices is called a…

A complete graph is the graph in which every two vertices are adjacent. For a graph $G=(V,E)$, the complete width of $G$ is the minimum $k$ such that there exist $k$ independent sets $\mathtt{N}_i\subseteq V$, $1\le i\le k$, such that the…

In this paper, we show that every planar graph without $4$-cycles and $6$-cycles has a partition of its vertex set into two sets, where one set induces a forest, and the other induces a forest with maximum degree at most $2$ (equivalently,…

Let $k_r(n,\delta)$ be the minimum number of $r$-cliques in graphs with $n$ vertices and minimum degree $\delta$. We evaluate $k_r(n,\delta)$ for $\delta \leq 4n/5$ and some other cases. Moreover, we give a construction, which we conjecture…

Every triangle-free planar graph on n vertices has an independent set of size at least (n+1)/3, and this lower bound is tight. We give an algorithm that, given a triangle-free planar graph G on n vertices and an integer k>=0, decides…

We show that the size of maximum cut in a planar graph with $m$ edges is at least $2m/3$. We also show that maximal planar graphs saturate this bound.

The independence number of a hypergraph H is the size of a largest set of vertices containing no edge of H. In this paper, we prove new sharp bounds on the independence number of n-vertex (r+1)-uniform hypergraphs in which every r-element…

The problem of determining the maximum number of copies of $T$ in an $H$-free graph, for any graphs $T$ and $H$, was considered by Alon and Shikhelman. This is a variant of Tur\'{a}n's classical extremal problem. We show lower and upper…