Let us consider a~simple connected undirected graph \(G=(V,E)\). For a~graph \(G\) we define a~\(k\)-labeling \(\phi: V(G)\to \{1,2, \dots, k\}\) to be a~distance irregular vertex \(k\)-labeling of the graph \(G\) if for every two different vertices \(u\) and \(v\) of \(G\), one has \(wt(u) \ne wt(v),\) where the weight of a~vertex \(u\) in the labeling \(\phi\) is \(wt(u)=\sum\limits_{v\in N(u)}\phi(v),\) where \(N(u)\) is the set of neighbors of \(u\). The minimum \(k\) for which the graph \(G\) has a~distance irregular vertex \(k\)-labeling is known as distance irregularity strength of \(G,\) it is denoted as \(dis(G)\). In this paper, we determine the exact value of the distance irregularity strength of corona product of cycle and path with complete graph of order \(1,\) friendship graph, Jahangir graph and helm graph. For future research, we suggest some open problems for researchers of the same domain of study.
Elimination ideals are monomial ideals associated to simple graphs, not necessarily square–free, was introduced by Anwar and Khalid. These ideals are Borel type. In this paper, we obtain sharp combinatorial upper bounds of the Castelnuovo–Mumford regularity of elimination ideals corresponding to certain family of graphs.
Let \(G\) be a simple connected graph with vertex set \(V\) and diameter \(d\). An injective function \(c: V\rightarrow \{1,2,3,\ldots\}\) is called a radio labeling of \(G\) if \({|c(x) c(y)|+d(x,y)\geq d+1}\) for all distinct \(x,y\in V\), where \(d(x,y)\) is the distance between vertices \(x\) and \(y\). The largest number in the range of \(c\) is called the span of the labeling \(c\). The radio number of \(G\) is the minimum span taken over all radio labelings of \(G\). For a fixed vertex \(z\) of \(G\), the sequence \((l_1,l_2,\ldots,l_r)\) is called the level tuple of \(G\), where \(l_i\) is the number of vertices whose distance from \(z\) is \(i\). Let\(J^k(l_1,l_2,\ldots,l_r)\) be the wedge sum (i.e. one vertex union) of \(k\geq2\) graphs having same level tuple \((l_1,l_2,\ldots,l_r)\). Let \(J(\frac{l_1}{l’_1},\frac{l_2}{l’_2},\ldots,\frac{l_r} {l’_r})\) be the wedge sum of two graphs of same order, having level tuples \((l_1,l_2,\ldots,l_r)\) and \((l’_1,l’_2,\ldots,l’_r)\). In this paper, we compute the radio number for some sub-families of \(J^k(l_1,l_2,\ldots,l_r)\) and \(J(\frac{l_1}{l’_1},\frac{l_2}{l’_2},\ldots,\frac{l_r}{l’_r})\).
An antipodal labeling is a function \(f\ \)from the vertices of \(G\) to the set of natural numbers such that it satisfies the condition \(d(u,v) + \left| f(u) – f(v) \right| \geq d\), where d is the diameter of \(G\ \)and \(d(u,v)\) is the shortest distance between every pair of distinct vertices \(u\) and \(v\) of \(G.\) The span of an antipodal labeling \(f\ \)is \(sp(f) = \max\{|f(u) – \ f\ (v)|:u,\ v\, \in \, V(G)\}.\) The antipodal number of~G, denoted by~an(G), is the minimum span of all antipodal labeling of~G. In this paper, we determine the antipodal number of Mongolian tent and Torus grid.
Let \( G \) be a \((p,q)\)-graph in which the edges are labeled \( k, k+1, \ldots, k+q-1 \), where \( k \geq 0 \). The vertex sum for a vertex \( v \) is the sum of the labels of the incident edges at \( v \). If the vertex sums are constant, modulo \( p \), then \( G \) is said to be \( k \)-edge-magic. In this paper, we investigate some classes of cubic graphs which are \( k \)-edge-magic. We also provide a counterexample to a conjecture that any cubic graph of order \( p \equiv 2 \pmod{4} \) is \( k \)-edge-magic for all \( k \).
In this paper, we prove the existence of \(22\) new \(3\)-designs on \(26\) and \(28\) points. The base of the constructions are two designs with a small maximum size of the intersection of any two blocks.
A large set of KTS(\(v\)), denoted by LKTS(\(v\)), is a collection of (\(v-2\)) pairwise disjoint KTS(\(v\)) on the same set. In this article, some new LKTS(\(v\)) is constructed.
Let \(G\) be a graph with \(v\) vertices. If there exists a list of colors \(S_1, S_2, \ldots, S_v\) on its vertices, each of size \(k\), such that there exists a unique proper coloring for \(G\) from this list of colors, then \(G\) is called a uniquely \(k\)-list colorable graph. We prove that a connected graph is uniquely \(2\)-list colorable if and only if at least one of its blocks is not a cycle, a complete graph, or a complete bipartite graph. For each \(k\), a uniquely \(k\)-list colorable graph is introduced.
A supergraph \(H\) of a graph \(G\) is called tree-covered if \(H – E(G)\) consists of exactly \(|V(G)|\) vertex-disjoint trees, with each tree having exactly one point in common with \(G\). In this paper, we show that if a graph \(G\) can be packed in its complement and if \(H\) is a tree-covered supergraph of \(G\), then \(G\) itself is self-packing unless \(H\) happens to be a member of a specified class of graphs. This is a generalization of earlier results that almost all trees and unicyclic graphs can be packed in their complements.
Let \(T = (V,A)\) be an oriented graph with \(n\) vertices. \(T\) is completely strong path-connected if for each arc \((a,b) \in A\) and \(k\) (\(k = 2, \ldots, n-1\)), there is a path from \(b\) to \(a\) of length \(k\) (denoted by \(P_k(a,b)\)) and a path from \(a\) to \(b\) of length \(k\) (denoted by \(P’_k(a,b)\)) in \(T\). In this paper, we prove that a connected local tournament \(T\) is completely strong path-connected if and only if for each arc \((a,b) \in A\), there exist \(P_2(a,b)\) and \(P’ _2(a,b)\) in \(T\), and \(T\) is not of \(T_1 \ncong T_0\)-\(D’_8\)-type digraph and \(D_8\).