Related papers: Cordial Digraphs
We generalize the class of split graphs to the directed case and show that these split digraphs can be identified from their degree sequences. The first degree sequence characterization is an extension of the concept of splittance to…
The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex (edge) labeling with $d$ labels that is preserved only by the trivial automorphism. It is known that for every graph $G$…
An acyclic digraph in which every vertex has indegree at most $i$ and outdegree at most $j$ is called an $(i,j)$ digraph for some positive integers $i$ and $j$. The phylogeny graph of a digraph $D$ has $V(D)$ as the vertex set and an edge…
A cyclic base ordering of a connected graph $G$, is a cyclic ordering of $E(G)$ such that every cyclically consecutive $|V(G)|-1$ edges form a spanning tree. In this project, we study cyclic base ordering of various families of graphs,…
A temporal graph is a graph in which edges are assigned a time label. Two nodes u and v of a temporal graph are connected one to the other if there exists a path from u to v with increasing edge time labels. We consider the problem of…
Which conditions ensure that a digraph contains all oriented paths of some given length, or even a all oriented trees of some given size, as a subgraph? One possible condition could be that the host digraph is a tournament of a certain…
Contraction of an edge merges its end points into a new vertex which is adjacent to each neighbor of the end points of the edge. An edge in a $k$-connected graph is {\em contractible} if its contraction does not result in a graph of lower…
A consistent path system in a graph $G$ is an intersection-closed collection of paths, with exactly one path between any two vertices in $G$. We call $G$ metrizable if every consistent path system in it is the system of geodesic paths…
In this paper, controllability of systems defined on graphs is discussed. We consider the problem of controllability of the network for a family of matrices carrying the structure of an underlying directed graph. A one-to-one correspondence…
For a graph $G$, let $\sigma_{2}(G)$ be the minimum degree sum of two non-adjacent vertices in $G$. A chord of a cycle in a graph $G$ is an edge of $G$ joining two non-consecutive vertices of the cycle. In this paper, we prove the following…
A prime labeling of a graph of order $n$ is a labeling of the vertices with the integers $1$ to~$n$ in which adjacent vertices have relatively prime labels. A coprime labeling maintains the same criterion on adjacent vertices using any set…
A graph is closed when its vertices have a labeling by $[n]$ such that the binomial edge ideal $J_G$ has a quadratic Gr\"{o}bner basis with respect to the lexicographic order induced by $x_1 > \cdots > x_n > y_1> \cdots > y_n$. In this…
A graph $G$ is called edge-magic if there is a bijective function $f$ from the set of vertices and edges to the set $\{1,2,\ldots,|V(G)|+|E(G)|\}$ such that the sum $f(x)+f(xy)+f(y)$ for any $xy$ in $E(G)$ is constant. Such a function is…
One model of message delivery in a computer network is based on labelling each edge by a subset of a (reasonably small) universal set, and then encoding a path as the union of the labels of its edges. Earlier work suggested using random…
A digraph $D$ is an oriented graph if $D$ does not have a pair of opposite arcs. The degree of a vertex $v$ of $D$ is the sum of the in-degree and out-degree of $v.$ Let $fvs(D)$ be the minimum number of vertices whose deletion from $D$…
A sequence $D=(d_1,d_2,\ldots,d_n)$ of non-negative integers is called a graphic sequence if there is a simple graph with vertices $v_1,v_2,\ldots,v_n$ such that the degree of $v_i$ is $d_i$ for $1\leq i\leq n$. Given a graph theoretical…
Given a digraph, an ordering of its vertices defines a backedge graph, namely the undirected graph whose edges correspond to the arcs pointing backwards with respect to the order. The degreewidth of a digraph is the minimum over all…
The $k$-coprime graph of order $n$ is the graph with vertex set $\{k, k+1, \ldots, k+n-1\}$ in which two vertices are adjacent if and only if they are coprime. We characterize Hamiltonian $k$-coprime graphs. As a particular case, two…
An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of $+1$ or $-1$. The adjacency and Laplacian eigenvalues of an oriented hypergraph are studied. Eigenvalue bounds for both the adjacency and Laplacian…
A graph is subcubic if it is connected and its maximum vertex degree does not exceed 3. Two disjoint vertex subsets of a graph $G$ form a connected coalition in $G$ if neither of them is a connected dominating set but their union is a…