Related papers: On density conditions for transversal trees in mul…
We prove that if a tree $T$ has $n$ vertices and maximum degree at most $\Delta$, then a copy of $T$ can almost surely be found in the random graph $\mathcal{G}(n,\Delta\log^5 n/n)$.
In 1970 Lov{\'a}sz gave a necessary and sufficient condition for the existence of a factor $F$ in a graph $G$ such that for each vertex $v$, $g(v)\le d_F(v)\le f(v)$, where $g$ and $f$ are two integer-valued functions on $V(G)$ with $g\le…
In this paper, we investigate the structural properties of trees and bipartite graphs through the lens of topological indices and combinatorial graph theory. We focus on the First and Second Hyper-Zagreb indices, $HM_1(G)$ and $HM_2(G)$,…
A random intersection graph is constructed by independently assigning a subset of a given set of objects $W,$ to each vertex of the vertex set $V$ of a simple graph $G.$ There is an edge between two vertices of $V,$ iff their respective…
We propose a random bipartite graph with weights assigned to both parts of the vertex sets. Edges are formed independently with probabilities that depend on these weights. This bipartite graph naturally gives rise to a random intersection…
A recent result of Condon, Kim, K\"{u}hn and Osthus implies that for any $r\geq (\frac{1}{2}+o(1))n$, an $n$-vertex almost $r$-regular graph $G$ has an approximate decomposition into any collections of $n$-vertex bounded degree trees. In…
We investigate the threshold probability for connectivity of sparse graphs under weak assumptions. As a corollary this completely solve the problem for Cartesian powers of arbitrary graphs. In detail, let $G$ be a connected graph on $k$…
The stability number of a graph G is the cardinality of a stability system of G (that is of a stable set of maximum size of G). A graph is alpha-stable if its stability number remains the same upon both the deletion and the addition of any…
Consider a drawing of a graph $G$ in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of $G$, is the classical graph parameter "thickness". By restricting the edges to be…
A $(q,r)$\emph{-tree-coloring} of a graph $G$ is a $q$-coloring of vertices of $G$ such that the subgraph induced by each color class is a forest of maximum degree at most $r.$ An \emph{equitable $(q, r)$-tree-coloring} of a graph $G$ is a…
A \emph{tree-partition} of a graph $G$ is a proper partition of its vertex set into `bags', such that identifying the vertices in each bag produces a forest. The \emph{tree-partition-width} of $G$ is the minimum number of vertices in a bag…
Given a system $\mathcal{G} =\{G_1,G_2,\dots,G_m\}$ of graphs/digraphs/hypergraphs on the common vertex set $V$ of size $n$, an $m$-edge graph/digraph/hypergraph $H$ on $V$ is transversal in $\mathcal{G}$ if there exists a bijection…
This paper establishes sufficient conditions that force a graph to contain a bipartite subgraph with a given structural property. In particular, let $\beta$ be any of the following graph parameters: Hadwiger number, Haj\'{o}s number,…
We consider a random geometric hypergraph model based on an underlying bipartite graph. Nodes and hyperedges are sampled uniformly in a domain, and a node is assigned to those hyperedges that lie with a certain radius. From a modelling…
Let $F$ be a graph which contains an edge whose deletion reduces its chromatic number. For such a graph $F,$ a classical result of Simonovits from 1966 shows that every graph on $n\ge n_0(F)$ vertices with more than…
We investigate the maximum size of graph families on a common vertex set of cardinality $n$ such that the symmetric difference of the edge sets of any two members of the family satisfies some prescribed condition. We solve the problem…
Let d_i(G) be the density of the 3-vertex i-edge graph in a graph G, i.e., the probability that three random vertices induce a subgraph with i edges. Let S be the set of all quadruples (d_0,d_1,d_2,d_3) that are arbitrary close to 3-vertex…
We characterize the bipartite graphs that minimize the (first-degree based) entropy, among all bipartite graphs of given size, or given size and (upper bound on the) order. The extremal graphs turn out to be complete bipartite graphs, or…
We determine asymptotically the two extremal constructions for the tiling problem of the $H$-shaped tree. In particular, the first extremal construction is close to the complement of two cliques, in contrast to previously studied bipartite…
With applications in distribution systems and communication networks, the minimum stretch spanning tree problem is to find a spanning tree T of a graph G such that the maximum distance in T between two adjacent vertices is minimized. The…