Related papers: On Integer Balancing of Digraphs
A $k$-edge-coloured graph is colour-balanced if each colour appears equally often. Resolving a conjecture of Pardey and Rautenbach, we show that any colour-balanced $k$-edge-coloured complete graph $K_{2kt}$ contains a perfect matching that…
A connected matching in a graph G consists of a set of pairwise disjoint edges whose covered vertices induce a connected subgraph of G. While finding a connected matching of maximum cardinality is a well-solved problem, it is NP-hard to…
Diophantine tuples are of ancient and modern interest, with a huge literature. In this paper we study Diophantine graphs, i.e., finite graphs whose vertices are distinct positive integers, and two vertices are linked by an edge if and only…
We analyze the threshold network model in which a pair of vertices with random weights are connected by an edge when the summation of the weights exceeds a threshold. We prove some convergence theorems and central limit theorems on the…
A graph labeling assigns values to the components of a graph (vertices, edges, etc.). In particular, distance magic labelings have been widely studied in undirected graphs. In such a labeling, the vertices are labeled with unique values…
Let $D$ be a weighted oriented graph and $I(D)$ be its edge ideal. If $D$ contains an induced odd cycle of length $2n+1$, under certain condition we show that $ {I(D)}^{(n+1)} \neq {I(D)}^{n+1}$. We give necessary and sufficient condition…
A d-partite hypergraph is called *fractionally balanced* if there exists a non-negative, not identically zero, function on its edge set that has constant degrees in each vertex side. Using a topological version of Hall's theorem we prove…
A graph $G$ on $n$ vertices with $k$ edges is $t$-edge-balanced if every graph on $n$ vertices with $t$ edges is contained in exactly the same number of subgraphs of $K_n$ isomorphic to $G$. Despite the existence of infinite families of…
For an arbitrary set of distances $D\subseteq \{0,1, \ldots, diam(G)\}$, a $D$-weight of a vertex $x$ in a graph $G$ under a vertex labeling $f:V\rightarrow \{1,2, \ldots , v\}$ is defined as $w_D(x)=\sum_{y\in N_D(x)} f(y)$, where $N_D(x)…
We characterise the form of all simple, finite graphs for which the girth of the graph is equal to the circumference of the graph. We apply this to prove a bound on the number of edges in such a graph.
A gain graph is a graph whose edges are orientably labelled from a group. A weighted gain graph is a gain graph with vertex weights from an abelian semigroup, where the gain group is lattice ordered and acts on the weight semigroup. For…
We classify all normal edge ideals of edge-weighted graphs.
An edge irregular total $k$-labelling $f : V(G)\cup E(G)\rightarrow \{1,2,\dots,k\}$ of a graph $G$ is a labelling of the vertices and the edges of $G$ in such a way that any two different edges have distinct weights. The weight of an edge…
In this paper, we study the componentwise linearity of edge ideals of weighted oriented graphs. We show that if $D$ is a weighted oriented graph whose edge ideal $I(D)$ is componentwise linear, then the underlying simple graph $G$ of $D$ is…
Let $G$ be a simple graph with $n$ vertices and $\pm 1$-weights on edges. Suppose that for every edge $e$ the sum of edges adjacent to $e$ (including $e$ itself) is positive. Then the sum of weights over edges of $G$ is at least…
Given a finite directed graph with $n$ vertices, we define a metric $d_G$ on $\mathbb{F}_q^n$, where $\mathbb{F}_q$ is the finite field with $q$ elements. The weight of a word is defined as the number of vertices that can be reached by a…
A k-digraph is an orientation of a multi-graph that is without loops and contains at most k edges between any pair of distinct vertices. We obtain necessary and sufficient conditions for a sequence of non-negative integers in non-decreasing…
An edge-coloring of a complete graph with a set of colors $C$ is called completely balanced if any vertex is incident to the same number of edges of each color from $C$. Erd\H{o}s and Tuza asked in $1993$ whether for any graph $F$ on $\ell$…
A graph is a mathematical object consisting of a set of vertices and a set of edges connecting vertices. Graphs can be drawn on paper in various ways, but until recently all published methods of drawing graphs have had undesirable…
Signed networks are graphs whose edges are labelled with either a positive or a negative sign, and can be used to capture nuances in interactions that are missed by their unsigned counterparts. The concept of balance in signed graph theory…