Authors:Stephen James Curran Abstract: It is known that for any prime p and any integer n such that 1≤n≤p there exists a prime labeling on the pxn planar grid graph PpxPn. We show that PpxPn has a prime labeling for any odd prime p and any integer n such that that p PubDate: Thu, 24 Mar 2022 07:06:37 PDT

Authors:Jana Coroničová Hurajová et al. Abstract: The \emph{betweenness centality} of an edge $e$ is, summed over all $u,v\in V(G)$, the ratio of the number of shortest $u,v$-paths in $G$ containing $e$ to the number of shortest $u,v$-paths in $G$. Graphs whose vertices all have the same edge betweenness centrality are called \emph{edge betweeness-uniform}. It was recently shown by Madaras, Hurajová, Newman, Miranda, Fl\'{o}rez, and Narayan that of the over 11.7 million graphs with ten vertices or fewer, only four graphs are edge betweenness-uniform but not edge-transitive.In this paper we present new results involving properties of betweenness-uniform graphs. PubDate: Thu, 24 Mar 2022 07:06:27 PDT

Authors:Deepak Sehrawat et al. Abstract: For $m \geq 3$ and $n \geq 1$, the $m$-cycle \textit{book graph} $B(m,n)$ consists of $n$ copies of the cycle $C_m$ with one common edge. In this paper, we prove that (a) the number of switching non-isomorphic signed $B(m,n)$ is $n+1$, and (b) the chromatic number of a signed $B(m,n)$ is either 2 or 3. We also obtain explicit formulas for the chromatic polynomials and the zero-free chromatic polynomials of switching non-isomorphic signed book graphs. PubDate: Thu, 24 Mar 2022 07:06:16 PDT

Authors:Richard M. Low et al. Abstract: Let $A$ be a nontrivial abelian group and $A^* = A \setminus \{0\}$. A graph is $A$-magic if there exists an edge labeling $f$ using elements of $A^*$ which induces a constant vertex labeling of the graph. Such a labeling $f$ is called an $A$-magic labeling and the constant value of the induced vertex labeling is called an $A$-magic value. In this paper, we use the Combinatorial Nullstellensatz to show the existence of $\mathbb{Z}_p$-magic labelings (prime $p \geq 3$ ) for various graphs, without having to construct the $\mathbb{Z}_p$-magic labelings. Through many examples, we illustrate the usefulness (and limitations) in applying the Combinatorial Nullstellensatz to the integer-magic labeling problem. Finally, we focus on $\mathbb{Z}_3$-magic labelings and give some results for various classes of graphs. PubDate: Tue, 22 Feb 2022 14:46:32 PST

Authors:Remie Janssen et al. Abstract: The unit distance graph $G^1_{R^d}$ is the infinite graph whose nodes are points in $R^d$, with an edge between two points if the Euclidean distance between these points is $1$. The 2-dimensional version $G^1_{R^2}$ of this graph is typically studied for its chromatic number, as in the Hadwiger-Nelson problem. However, other properties of unit distance graphs are rarely studied. Here, we consider the restriction of $G^1_{R^d}$ to closed convex subsets $X$ of $R^d$. We show that the graph $G^1_{R^d}[X]$ is connected precisely when the radius of $r(X)$ of $X$ is equal to $0$, or when $r(X)\geq 1$ and the affine dimension of $X$ is at least $2$. For hyperrectangles, we give bounds for the graph diameter in the critical case that the radius is exactly 1. PubDate: Tue, 22 Feb 2022 14:41:33 PST

Authors:Naoki Matsumoto et al. Abstract: A (not necessarily proper) $k$-coloring $c : V(G) \rightarrow \{1,2,\dots,k\}$ of a graph $G$ on a surface is a {\em facial $t$-complete $k$-coloring} if every $t$-tuple of colors appears on the boundary of some face of $G$. The maximum number $k$ such that $G$ has a facial $t$-complete $k$-coloring is called a {\em facial $t$-achromatic number} of $G$, denoted by $\psi_t(G)$. In this paper, we investigate the relation between the facial 3-achromatic number and guarding number of triangulations on a surface, where a {\em guarding number} of a graph $G$ embedded on a surface, denoted by $\gd(G)$, is the smallest size of its {\em guarding set} which is a generalized concept of guards in the art gallery problem. We show that for any graph $G$ embedded on a surface, $\psi_{\Delta(G^*)}(G) \leq \gd(G) + \Delta(G^*) - 1$, where $\Delta(G^*)$ is the largest face size of $G$. Furthermore, we investigate sufficient conditions for a triangulation $G$ on a surface to satisfy $\psi_{3}(G) = \gd(G) + 2$. In particular, we prove that every triangulation $G$ on the sphere with $\gd(G) = 2$ satisfies the above equality and that for one with guarding number $3$, it also satisfies the above equality with sufficiently large number of vertices. PubDate: Wed, 16 Feb 2022 09:26:17 PST

Authors:Willie Han Wah Wong et al. Abstract: For a connected graph $G$, let $\mathscr{D}(G)$ be the family of strong orientations of $G$; and for any $D\in\mathscr{D}(G)$, we denote by $d(D)$ the diameter of $D$. The $\textit{orientation number}$ of $G$ is defined as $\bar{d}(G)=\min\{d(D)\mid D\in \mathscr{D}(G)\}$. In 2000, Koh and Tay introduced a new family of graphs, $G$ vertex-multiplications, and extended the results on the orientation number of complete $n$-partite graphs. Suppose $G$ has the vertex set $V(G)=\{v_1,v_2,\ldots, v_n\}$. For any sequence of $n$ positive integers $(s_i)$, a $G$ \textit{vertex-multiplication}, denoted by $G(s_1, s_2,\ldots, s_n)$, is the graph with vertex set $V^*=\bigcup_{i=1}^n{V_i}$ and edge set $E^*$, where $V_i$'s are pairwise disjoint sets with $ V_i =s_i$, for $i=1,2,\ldots,n$; and for any $u,v\in V^*$, $uv\in E^*$ if and only if $u\in V_i$ and $v\in V_j$ for some $i,j\in \{1,2,\ldots, n\}$ with $i\neq j$ such that $v_i v_j\in E(G)$. They proved a fundamental classification of $G$ vertex-multiplications, with $s_i\ge 2$ for all $i=1,2,\ldots, n$, into three classes $\mathscr{C}_0, \mathscr{C}_1$ and $\mathscr{C}_2$, and any vertex-multiplication of a tree with diameter at least 3 does not belong to the class $\mathscr{C}_2$. Furthermore, some necessary and sufficient conditions for $\mathscr{C}_0$ were established for vertex-multiplications of trees with diameter $5$. In this paper, we give a complete characterisation of vertex-multiplications of trees with diameter $5$ in $\mathscr{C}_0$ and $\mathscr{C}_1$. PubDate: Mon, 15 Nov 2021 10:31:27 PST

Authors:Elliot Krop et al. Abstract: In any graph $G$, the domination number $\gamma(G)$ is at most the independence number $\alpha(G)$. The \emph{Inverse Domination Conjecture} says that, in any isolate-free $G$, there exists pair of vertex-disjoint dominating sets $D, D'$ with $ D =\gamma(G)$ and $ D' \leq \alpha(G)$. Here we prove that this statement is true if the upper bound $\alpha(G)$ is replaced by $\frac{3}{2}\alpha(G) - 1$ (and $G$ is not a clique). We also prove that the conjecture holds whenever $\gamma(G)\leq 5$ or $ V(G) \leq 16$. PubDate: Wed, 25 Aug 2021 09:42:18 PDT

Authors:Braxton Carrigan et al. Abstract: A Skolem sequence can be thought of as a labelled path where two vertices with the same label are that distance apart. This concept has naturally been generalized to labellings of other graphs, but always using at most two of any integer label. Given that more than two vertices can be mutually distance d apart, we define a new generalization of a Skolem sequences on graphs that we call proper Skolem labellings. This brings rise to the question; ``what is the smallest set of consecutive positive integers we can use to proper Skolem label a graph''' This will be known as the Skolem number of the graph. In this paper we give the Skolem number for cycles and grid graphs, while also providing other related results along the way. PubDate: Wed, 25 Aug 2021 09:42:05 PDT

Authors:Merlin Thomas Ellumkalayil Ms et al. Abstract: In a proper vertex colouring of a graph, the vertices are coloured in such a way that no two adjacent vertices receive the same colour, whereas in an improper vertex colouring, adjacent vertices are permitted to receive same colours subjected to some conditions. An edge of an improperly coloured graph is said to be a bad edge if its end vertices have the same colour. A near proper colouring is a colouring which minimises the number of bad edges by restricting the number of colour classes that can have adjacency among their own elements. The $\delta^{(k)}$- colouring is a near proper colouring of $G$ consisting of $k$ given colours, which minimises the number of bad edges by permitting at most one colour class to have adjacency among the vertices in it. In this paper, we determine the number of bad edges of powers of Paths $(P_{n})$ and powers of Cycles $(C_{n})$. PubDate: Wed, 18 Aug 2021 09:28:31 PDT

Authors:Jill Faudree et al. Abstract: An edge-colored graph is properly connected if for every pair of vertices u and v there exists a properly colored uv-path (i.e. a uv-path in which no two consecutive edges have the same color). The proper connection number of a connected graph G, denoted pc(G), is the smallest number of colors needed to color the edges of G such that the resulting colored graph is properly connected. An edge-colored graph is flexibly connected if for every pair of vertices u and v there exist two properly colored paths between them, say P and Q, such that the first edges of P and Q have different colors and the last edges of P and Q have different colors. The flexible connection number of a connected graph G, denoted fpc(G), is the smallest number of colors needed to color the edges of G such that the resulting colored graph is flexibly connected. In this paper, we demonstrate several methods for constructing graphs with pc(G) = 2 and fpc(G) = 2. We describe several families of graphs such that pc(G) ≥ 2 and we settle a conjecture from [BFG+12]. We prove that if G is connected and bipartite, then pc(G) = 2 is equivalent to being 2-edge-connected and fpc(G) = 2 is equivalent to the existence of a path through all cut-edges. Finally, it is proved that every connected, k-regular, Class 1 graph has flexible connection number 2. PubDate: Mon, 02 Aug 2021 07:26:14 PDT

Authors:Analen A. Malnegro et al. Abstract: The color number c(G) of a cubic graph G is the minimum cardinality of a color class of a proper 4-edge-coloring of G. It is well-known that every cubic graph G satisfies c(G) = 0 if G has a Hamiltonian cycle, and c(G) ≤ 2 if G has a Hamiltonian path. In this paper, we extend these observations by obtaining a bound for the color number of cubic graphs having a spanning tree with a bounded number of leaves. PubDate: Sat, 31 Jul 2021 14:02:50 PDT

Authors:Paweł Marcin Kozyra Abstract: The notion of a replica of a nontrivial in-tree is defined. A result enabling to determine whether an in-tree is a replica of another in-tree employing an injective mapping between some subsets of sources of these in-trees is presented. There are given necessary and sufficient conditions for the existence of a functional square root of a function from a finite set to itself through presenting necessary and sufficient conditions for the existence of a square root of a component of the functional graph for the function and for the existence of a square root of the union of two components of the functional graph for the function containing cycles of the same length using the concept of the replica. PubDate: Fri, 25 Jun 2021 04:56:17 PDT

Authors:Angsuman Das et al. Abstract: Rose window graphs are a family of tetravalent graphs, introduced by Steve Wilson. Following it, Kovacs, Kutnar and Marusic classified the edge-transitive rose window graphs and Dobson, Kovacs and Miklavic characterized the vertex transitive rose window graphs. In this paper, we classify the Cayley rose window graphs. PubDate: Tue, 27 Apr 2021 08:07:04 PDT

Authors:Peter Borg Abstract: Let $\lambda(G)$ be the smallest number of vertices that can be removed from a non-empty graph $G$ so that the resulting graph has a smaller maximum degree. Let $\lambda_{\rm e}(G)$ be the smallest number of edges that can be removed from $G$ for the same purpose. Let $k$ be the maximum degree of $G$, let $t$ be the number of vertices of degree $k$, let $M(G)$ be the set of vertices of degree $k$, let $n$ be the number of vertices in the closed neighbourhood of $M(G)$, and let $m$ be the number of edges that have at least one vertex in $M(G)$. Fenech and the author showed that $\lambda(G) \leq \frac{n+(k-1)t}{2k}$, and they essentially showed that $\lambda (G) \leq n \left ( 1- \frac{k}{k+1} { \Big( \frac{n}{(k+1)t} \Big) }^{1/k} \right )$. They also showed that $\lambda_{\rm e}(G) \leq \frac{m + (k-1)t}{2k-1}$ and that if $k \geq 2$, then $\lambda_{\rm e} (G) \leq m \left ( 1- \frac{k-1}{k} { \Big( \frac{m}{kt} \Big) }^{1/(k-1)} \right )$. These bounds are attained if $G$ is the union of pairwise vertex-disjoint $(k+1)$-vertex stars. In this paper, we determine the cases in which one bound on $\lambda(G)$ is better than the other, and we show that the first bound on $\lambda_{\rm e}(G)$ is better than the second. This work is motivated by the likelihood that similar pairs of bounds will be discovered for other graph parameters and the same analysis can be applied. PubDate: Mon, 12 Apr 2021 14:46:48 PDT

Authors:Mohamad Abdallah et al. Abstract: The strong matching preclusion is a measure for the robustness of interconnection networks in the presence of node and/or link failures. However, in the case of random link and/or node failures, it is unlikely to find all the faults incident and/or adjacent to the same vertex. This motivates Park et al. to introduce the conditional strong matching preclusion of a graph. In this paper we consider the conditional strong matching preclusion problem of the augmented cube $AQ_n$, which is a variation of the hypercube $Q_n$ that possesses favorable properties. PubDate: Thu, 25 Mar 2021 06:56:28 PDT

Authors:Terry A. McKee Abstract: The 2-connected 2-tree graphs are defined as being constructible from a single 3-cycle by recursively appending new degree-2 vertices so as to form 3-cycles that have unique edges in common with the existing graph. Such 2-trees can be characterized both as the edge-minimal chordal graphs and also as the edge-maximal series-parallel graphs. These are also precisely the 2-connected graphs that are simultaneously chordal and series-parallel, where these latter two better-known types of graphs have themselves been both characterized and applied in numerous ways that are unmotivated by their interaction with 2-trees and with each other.Toward providing such motivation, the present paper examines several simple, very closely-related characterizations of chordal graphs and 2-trees and, after that, of series-parallel graphs and 2-trees. This leads to showing a way in which chordal graphs and series-parallel graphs are special---indeed, extremal---in this regard. PubDate: Thu, 25 Mar 2021 06:21:21 PDT

Authors:Kai Wang Abstract: As a partial answer to a question of Rao, a deterministic and customizable efficient algorithm is presented to test whether an arbitrary graphical degree sequence has a bipartite realization. The algorithm can be configured to run in polynomial time, at the expense of possibly producing an erroneous output on some ``yes'' instances but with very low error rate. PubDate: Fri, 05 Mar 2021 13:46:58 PST

Authors:Dhiren Kumar Basnet et al. Abstract: In this article, we introduce the concept of nilpotent graph of a finite commutative ring. The set of all non nilpotent elements of a ring is taken as the vertex set and two vertices are adjacent if and only if their sum is nilpotent. We discuss some graph theoretic properties of nilpotent graph. PubDate: Thu, 18 Feb 2021 11:49:49 PST

Authors:Richard M. Low et al. Abstract: Let $A$ be a nontrival abelian group. A connected simple graph $G = (V, E)$ is $A$-antimagic if there exists an edge labeling $f: E(G) \to A \setminus \{0\}$ such that the induced vertex labeling $f^+: V(G) \to A$, defined by $f^+(v) = \sum_{uv\in E(G)}f(uv)$, is injective. The integer-antimagic spectrum of a graph $G$ is the set IAM$(G) = \{k\; \; G \textnormal{ is } \mathbb{Z}_k\textnormal{-antimagic}$ $\textnormal{and } k \geq 2\}$. In this paper, we determine the integer-antimagic spectra for cycles with a chord, paths with a chord, and wheels with a chord. PubDate: Tue, 09 Feb 2021 15:31:06 PST