Total outer-independent domination in regular graphs

Upper bounds for independent domination in regular graphs

Let G be a connected k(≥3)-regular graph with girth g. A set S of the edges in G is called an R2-edge-cut if G-S is disconnected and contains neither an isolated vertex nor a one-degree vertex. The R2-edge-connectivity of G, denoted by λn(G), is the minimum cardinality over all R2-edge-cuts, which is an important measure for fault-tolerance of computer interconnection networks. In this paper, λn(G)=g(2k−2) for any 2k-regular connected graph G(≠K5) that is either edge-transitive or vertex-transitive and g≥5 is given.

In a given graph G=(V, E), a set of vertices S with an assignment of colours to them is said to be a defining set of the vertex colouring of G, if there exists a unique extension of the colours of S to a c≥χ(G) colouring of the vertices of G. A defining set with minimum cardinality is called a minimum defining set and its cardinality is the defining number, denoted by d(G, c). The defining set S is strong, if there exists an ordering {v 1, v 2, …, v n–s } of the vertices of ⟨G–S⟩ such that in the induced list of colours in each of the subgraphs ⟨G–S⟩, ⟨G–S∪{v1}⟩, ⟨G–S∪{v 1, v 2}⟩ and ⟨G–S∪{v 1, v 2, …, v n–s }⟩, there exist at least one vertex whose list of colours is of cardinality 1. The strong defining number, sd(G, c) of G is the cardinality of its smallest strong defining set. If is a family of graphs then . Here we study the cases where is the family of k-regular (connected and disconnected) graphs on n vertices and c=k–1. Also the Speck–1() defining spectrum of all k-regular (connected and disconnected) graph on n vertices are verified for k=3 and 4.

An antimagic labeling of a graph G is a bijection f:E(G)→{1,…,|E(G)|} such that the weights w(x)=∑y∼xf(y) distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1990) is that every connected graph other than K2 admits an antimagic labeling. For a set of distances D, a D-antimagic labeling of a graph G is a bijection f:V(G)→{1,…,|V(G)|} such that the weightω(x)=∑y∈ND(x)f(y) is distinct for each vertex x, where ND(x)={y∈V(G)|d(x,y)∈D} is the D-neigbourhood set of a vertex x. If ND(x)=r, for every vertex x in G, a graph G is said to be (D,r)-regular. In this paper, we conjecture that a graph admits a D-antimagic labeling if and only if it does not contain two vertices having the same D-neighborhood set. We also provide evidence that the conjecture is true. We present computational results that, for D={1}, all graphs of order up to 8 concur with the conjecture. We prove that the set of (D,r)-regular D-antimagic graphs is closed under union. We provide examples of disjoint union of symmetric (D,r)-regular that are D-antimagic and examples of disjoint union of non-symmetric non-(D,r)-regular graphs that are D-antimagic. Furthermore, lastly, we show that it is possible to obtain a D-antimagic graph from a previously known distance antimagic graph.

Vertex-magic labelings of regular graphs II

Let [Formula: see text] and [Formula: see text] be positive integers, and let [Formula: see text] be a simple and connected graph with vertex set [Formula: see text]. For [Formula: see text], let [Formula: see text] denote the length of an [Formula: see text] geodesic in [Formula: see text] and let [Formula: see text]. Let [Formula: see text]. A set [Formula: see text] is a distance-[Formula: see text] locating set of [Formula: see text] if, for any distinct [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text], and the distance-[Formula: see text] location number, [Formula: see text], of [Formula: see text] is the minimum cardinality among all distance-[Formula: see text] locating sets of [Formula: see text]. A set [Formula: see text] is a distance-[Formula: see text] dominating set of [Formula: see text] if, for each vertex [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text], and the distance-[Formula: see text] domination number, [Formula: see text], of [Formula: see text] is the minimum cardinality among all distance-[Formula: see text] dominating sets of [Formula: see text]. The [Formula: see text]-location-domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality among all sets [Formula: see text] such that [Formula: see text] is both a distance-[Formula: see text] locating set and a distance-[Formula: see text] dominating set of [Formula: see text]. For any connected graph [Formula: see text] of order at least two, we show that [Formula: see text], where [Formula: see text]. We characterize connected graphs [Formula: see text] satisfying [Formula: see text] equals [Formula: see text] and [Formula: see text], respectively. We examine the relationship among [Formula: see text], [Formula: see text] and [Formula: see text]; along the way, we show that [Formula: see text] if [Formula: see text]. We also show that there exist graphs [Formula: see text] and [Formula: see text] with [Formula: see text] such that [Formula: see text] can be arbitrarily large. Moreover, we examine some graph classes.

Hamiltonian Cycles in Regular Graphs

A 1-factorisation of a graph G is a decomposition of G into edge-disjoint 1-factors (1-regular spanning subgraphs). A perfect 1-factorisation is a 1-factorisation in which the union of every pair of distinct 1-factors is a Hamilton cycle; a 1-factorisation is uniform if the graphs resulting from the union of every pair of distinct 1-factors are isomorphic. We initiate the study of perfect 1-factorisations and uniform 1-factorisations of circulant graphs, a well-studied family of Cayley graphs, by determining whether or not such 1-factorisations exist for even order circulant graphs of small degree. For a bipartite circulant graph, we observe that having order congruent to 2 (mod 4) is a necessary condition for the existence of a perfect 1-factorisation. We prove that for bipartite 3-regular circulant graphs this necessary condition is also sufficient. Furthermore, the bipartite 3-regular circulant graphs of order 2 (mod 4) are the only 3-regular circulant graphs of order greater than 6 which admit a perfect 1-factorisation. The 4-regular case is more complex; we provide constructions of perfect 1-factorisations for many families of bipartite 4-regular circulant graphs of order 2 (mod 4); however, we also prove that there is an infinite family of such graphs no member of which admits a perfect 1-factorisation. We conjecture that a connected non-bipartite 4-regular circulant graph of order at least 8 does not admit a perfect 1-factorisation; we prove this conjecture for three distinct infinite families of non-bipartite 4-regular circulant graphs and show by computer search that the conjecture holds for small orders (at most 30). Several of our results are achieved by studying perfect 1-factorisations of a particular family of 4-regular graphs, denoted Dh, k for integers h, knwhere h g 2 and k g 3. These graphs are shown to be isomorphic to Cayley graphs when k is even and yet they are not vertex-transitive when k is odd (and hng 3). For small values of h, we make use of the structure of Dh,k to determine when it admits a perfect 1-factorisation and then use specific isomorphism results to obtain corresponding results for 4-regular circulant graphs. In this thesis the connected 3-regular circulant graphs that admit a perfect 1-factorisation are characterised and although the connected 4-regular circulant graphs that admit a perfect 1-factorisation have not yet been fully characterised, we provide many interesting theoretical results that establish the framework for such a characterisation.n

Constructing an Infinite Family of Cubic 1-Regular Graphs

In this paper it is shown that if a graph G possesses a spanning subgraph H with a strong vertex magic total labeling (VMTL) and $G-E(H)$ is even‐regular, then G also has a strong VMTL. Among other things, this is used to conclude that all Hamiltonian regular graphs of odd order possess strong VMTLs. A relationship is then demonstrated between regular graphs of even degree and sparse magic squares. We next consider cubic graphs of order $2n$ consisting of two 2‐factors of order n, connected by a 1‐factor (quasi‐prisms). Based on McQuillan’s construction of VMTLs of such 3‐regular graphs, VMTLs are derived for similar regular graphs of any odd degree. Finally, a construction is given for VMTLs of quartic graphs of order $4n+2$ consisting of two cycles of odd order n connected by a 2‐factor (simple quasi‐anti‐prisms), and based on this construction VMTLs are derived for similar regular graphs of any even degree.

Least domination in a graph

Let G be a connected graph. Dominating set is a set of vertices which each vertex D has at least one neighbor in G. The minimum cardinality of D is called the domination number G(γ(G)). The metric dimension of G is the minimum cardinality of a series of vertices so that each vertex G is uniquely. It is determined by the distance of vector to the selected vertices. A dominating metric dimension set is a set of vertices has a dominating set D which has condition of metric dimension. The minimum cardinality is called the resolving domination number of G, (DomDim (G)). We analyze the resolving domination number of helm graph and it’s operation. We study combine the existence concept of dominating set and metric dimension. We have obtained the minimum cardinality of dominating number.

Let [Formula: see text] be a positive integer and [Formula: see text] be a simple connected graph. The eccentric distance sum of [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the maximum distance from [Formula: see text] to any other vertex and [Formula: see text] is the sum of all distances from [Formula: see text]. A set [Formula: see text] is a distance [Formula: see text]-dominating set of [Formula: see text] if for every vertex [Formula: see text], [Formula: see text] for some vertex [Formula: see text]. The minimum cardinality among all distance [Formula: see text]-dominating sets of [Formula: see text] is called the distance [Formula: see text]-domination number [Formula: see text] of [Formula: see text]. In this paper, the trees among all [Formula: see text]-vertex trees with distance [Formula: see text]-domination number [Formula: see text] having the minimal eccentric distance sum are determined.

A subset $X$ of edges of a graph $G$ is called an \textit{edge dominating set} of $G$ if every edge not in $X$ is adjacent to some edge in $X$. The edge domination number $\gamma'(G)$ of $G$ is the minimum cardinality taken over all edge dominating sets of $G$. An \textit{edge Roman dominating function} of a graph $G$ is a function $f : E(G)\rightarrow \{0,1,2 \}$ such that every edge $e$ with $f(e)=0$ is adjacent to some edge $e'$ with $f(e') = 2.$ The weight of an edge Roman dominating function $f$ is the value $w(f)=\sum_{e\in E(G)}f(e)$. The edge Roman domination number of $G$, denoted by $\gamma_R'(G)$, is the minimum weight of an edge Roman dominating function of $G$. In this paper, we characterize trees with edge Roman domination number twice the edge domination number.

For a positive integer $k$, a subset $D$ of vertices in a digraph $\overrightarrow{G}$ is a $k$-dominating set if every vertex not in $D$ has at least $k$ direct predecessors in $D.$ The $k$-domination number is the minimum cardinality among all $k$-dominating sets of $\overrightarrow{G}$. The game $k$-domination number of a simple and undirected graph is defined by the following game. Two players, $\mathcal{A}$ and $\mathcal{D}$, orient the edges of the graph alternately until all edges are oriented. Player $\mathcal{D}$ starts the game, and his goal is to decrease the $k$-domination number of the resulting digraph, while $\mathcal{A}$ is trying to increase it. The game $k$-domination number of the graph $G$ is the $k$-domination number of the directed graph resulting from this game. This is well defined if we suppose that both players follow their optimal strateries. We are mainly interested in the study of the game $2$-domination number, where some upper bounds will be presented. We also establish a Nordhaus-Gaddum bound for the game $2$-domination number of a graph and its complement.