Signed star (k,k)-domatic number of a graph

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A Roman k-dominating function on G is a function f : V (G) <TEX>$\rightarrow$</TEX> {0, 1, 2} such that every vertex u for which f(u) = 0 is adjacent to at least k vertices <TEX>$\upsilon_1,\;\upsilon_2,\;{\ldots},\;\upsilon_k$</TEX> with <TEX>$f(\upsilon_i)$</TEX> = 2 for i = 1, 2, <TEX>$\ldot$</TEX>, k. The weight of a Roman k-dominating function is the value f(V (G)) = <TEX>$\sum_{u{\in}v(G)}$</TEX> f(u). The minimum weight of a Roman k-dominating function on a graph G is called the Roman k-domination number <TEX>${\gamma}_{kR}$</TEX>(G) of G. Note that the Roman 1-domination number <TEX>$\gamma_{1R}$</TEX>(G) is the usual Roman domination number <TEX>$\gamma_R$</TEX>(G). In this paper, we investigate the properties of the Roman k-domination number. Some of our results extend these one given by Cockayne, Dreyer Jr., S. M. Hedetniemi, and S. T. Hedetniemi [2] in 2004 for the Roman domination number.

Let D be a finite and simple digraph with vertex set V(D), and let f: V(D) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and \({\sum_{x \in N^-(v)}f(x) \ge k}\) for each \({v \in V(G)}\), where N −(v) consists of all vertices of D from which arcs go into v, then f is a signed total k-dominating function on D. A set {f 1, f 2, . . . , f d } of signed total k-dominating functions on D with the property that \({\sum_{i=1}^df_i(x)\le k}\) for each \({x \in V(D)}\) , is called a signed total (k, k)-dominating family (of functions) on D. The maximum number of functions in a signed total (k, k)-dominating family on D is the signed total (k, k)-domatic number on D, denoted by \({d_{st}^{k}(D)}\) . In this paper we initiate the study of the signed total (k, k)-domatic number of digraphs, and we present different bounds on \({d_{st}^{k}(D)}\) . Some of our results are extensions of known properties of the signed total domatic number \({d_{st}(D)=d_{st}^{1}(D)}\) of digraphs D as well as the signed total domatic number d st (G) of graphs G, given by Henning (Ars Combin. 79:277–288, 2006).

Total labellings, irregularity strengths and colourings

The distance Roman domatic number of a graph

Let G be a simple graph with vertex set V(G) and edge set E(G). A vertex coloring of G is called a star coloring of G if any of the paths of 4 order are bicolored. The minimum number of colors required for a star coloring of G is denoted by χs (G). The corona product of simple graphs G of order m and H of order n is graph G ∘ H with vertex set V(G ∘ H) = {vi |i = 1,2,⋯m}∪{vij |i = 1,2,⋯m, j = 1,2,⋯n}, in which vi is adjacent to every vertex of Hi if and only if, vi ∈ V(G), vij ∈ V(Hi ). According to the existing graph dyeing literature, it has become a very important technical means to study the graph dyeing problem by using the graph structure operation. Therefore, it is of great significance to study the star coloring of graphs for studying the acyclic coloring and distance coloring of graphs, the study has strong application background and great theoretical value for computing graphs. In this paper, we find the upper bound of χs (G ∘ H) and the exact values of χs (G ∘ H) of the corona product G ∘ H of two graphs G and H as: χs (G ∘ H) ≤ χs (G) + χs (H); χs (Pm ∘ H) = χs (H) + 2; χs (K 1,m ∘ H) = χs (H) + 2; χs (Cn ∘ H) = χs (H) + 2, where n ≠ 5.

Let G = (V, E) be a finite and simple graph with vertex set V (G) and edge set E(G), having order p and size q. A graph G is called super edge-magic if there exists a bijection f : V(G) ∪ E(G) → {1, 2, …, p + q} such that f (V(G)) = {1, 2, …, p} and f (u) + f (uv) + f (v) = k, for every edge uv ∈ E(G). The super edge-magic deficiency of a graph G, denoted by μs(G), is either the minimum nonnegative integer n such that G ∪ nK1 is super edge-magic or +∞ if there exists no such n. In this paper, we study the super edge-magic deficiency of forests where its components are subdivided stars or paths.

Let G be a simple graph with vertex set V(G) and edge set E(G). Automorphism of G is an isomorphism from G to itself. In other words, an automorphism of G is a permutation φ of V(G), which has the property that uv ∈ E(G) if and only if φ(u)φ(v) ∈ E(G), for every u, v ∈ V(G) that is φ preserves adjacency. The complete n-ary tree is a rooted-tree where its every internal vertex has exactly n children. The height of a rooted tree is the number of edges on longest path connecting the root to a leaf (a vertex of degree one). In this paper, we determine the number of automorphism of complete 2,3,4-ary trees with height h, for any positive integer h. We also conjecture the general formula to determine the number of automorphism of complete n-ary trees, for any n ≥ 5.

A double Roman dominating function on a graph $G$ with vertex set $V(G)$ is defined in \cite{bhh} as a function‎ ‎$f:V(G)\rightarrow\{0,1,2,3\}$ having the property that if $f(v)=0$‎, ‎then the vertex $v$ must have at least two‎ ‎neighbors assigned 2 under $f$ or one neighbor $w$ with $f(w)=3$‎, ‎and if $f(v)=1$‎, ‎then the vertex $v$ must have‎ ‎at least one neighbor $u$ with $f(u)\ge 2$‎. ‎The weight of a double Roman dominating function $f$ is the sum‎ ‎$\sum_{v\in V(G)}f(v)$‎, ‎and the minimum weight of a double Roman dominating function on $G$ is the double Roman‎ ‎domination number $\gamma_{dR}(G)$ of $G$‎. ‎A set $\{f_1,f_2,\ldots,f_d\}$ of distinct double Roman dominating functions on $G$ with the property that‎ ‎$\sum_{i=1}^df_i(v)\le 3$ for each $v\in V(G)$ is called in \cite{v} a double Roman dominating family (of functions)‎ ‎on $G$‎. ‎The maximum number of functions in a double Roman dominating family on $G$ is the double Roman domatic number‎ ‎of $G$‎. ‎In this note we continue the study of the double Roman domination and domatic numbers‎. ‎In particular‎, ‎we present‎ ‎a sharp lower bound on $\gamma_{dR}(G)$‎, ‎and we determine the double Roman domination and domatic numbers of some‎ ‎classes of graphs.

Let \(D\) be a simple digraph with vertex set \(V(D)\), and let \(f:V(D)\rightarrow \{-1,1\}\) be a two-valued function. If \(k\ge 1\) is an integer and \(\sum _{x\in N^-[v]}f(x)\ge k\) for each \(v\in V(D)\), where \(N^-[v]\) consists of \(v\) and all vertices of \(D\) from which arcs go into \(v\), then \(f\) is a signed \(k\)-dominating function on \(D\). A set \(\{f_1,f_2,\ldots ,f_d\}\) of distinct signed \(k\)-dominating functions on \(D\) with the property that \(\sum _{i=1}^df_i(x)\le k\) for each \(x\in V(D)\), is called a signed \((k,k)\)-dominating family (of functions) on \(D\). The maximum number of functions in a signed \((k,k)\)-dominating family on \(D\) is the signed \((k,k)\)-domatic number of \(D\). In this article, we mainly present upper bounds on the signed \((k,k)\)-domatic number, in particular for regular digraphs.

A graph in this paper is nontrivial, finite, connected, simple, and undirected. Graph consists of a vertex set and edge set. Let u,v be two elements in vertex set, and q is the cardinality of edge set in G, a bijective function from the edge set to the first q natural number is called a vertex local antimagic edge labelling if for any two adjacent vertices and , the weight of is not equal with the weight of , where the weight of (denoted by ) is the sum of labels of edges that are incident to . Furthermore, any vertex local antimagic edge labelling induces a proper vertex colouring on where is the colour on the vertex . The vertex local antimagic chromatic number is the minimum number of colours taken over all colourings induced by vertex local antimagic edge labelling of . In this paper, we discuss about the vertex local antimagic chromatic number on disjoint union of some family graphs, namely path, cycle, star, and friendship, and also determine the lower bound of vertex local antimagic chromatic number of disjoint union graphs. The chromatic numbers of disjoint union graph in this paper attend the lower bound.

A (p,1)- total labeling of G is an assignment of integers to V(G)∪E(G) such that any two adjacent vertices of G receive distinct integers, any two adjacent edges of G receive distinct integers, and a vertex and its incident edge receive integers that differ by at least p in absolute value. The span of a (p,1)-total labeling is the maximum difference between two labels. The minimum span of a (p,1)-total labeling of G is called the (p,1)-total number and denoted by λ^T_P(G). Let n and k be two positive integers. The graph with vertex sets {u(1),...,u(n)} and {v(1),...,v(n)} and edge sets {u(i)u(i+1)|i=1,2,...,n},{u(i)v(i)|i=1,2,...,n} and {v(i)v(i+k)|i=1,2,...,n;k<n}, where addition is modulo n is called generalized Petersen graph and denoted by P(n,k). In this thesis, we mainly focus on the (2,1)-total labeling of the generalized Petersen graph, and we show that for each pair of positive integer n and k, 1<=n, if n≡0(mod 5) and k≠0(mod 5), then λ^T_2(P(n,k))=5. Moreover, we also prove that λ^T_2(P(n,5))=5 if n≡0(mod 25).

Minimum face-spanning subgraphs of plane graphs

Let G(V, E) be a simple, undirected graph where V is the set of vertices and E is the set of edges. A b‐dimensional cube is a Cartesian product I1×I2×···×Ib, where each Ii is a closed interval of unit length on the real line. The cubicity of G, denoted by cub(G), is the minimum positive integer b such that the vertices in G can be mapped to axis parallel b‐dimensional cubes in such a way that two vertices are adjacent in G if and only if their assigned cubes intersect. An interval graph is a graph that can be represented as the intersection of intervals on the real line—i.e. the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. Suppose S(m) denotes a star graph on m+1 nodes. We define claw number ψ(G) of the graph to be the largest positive integer m such that S(m) is an induced subgraph of G. It can be easily shown that the cubicity of any graph is at least ⌈log2ψ(G)⌉. In this article, we show that for an interval graph G ⌈log2ψ(G)⌉⩽cub(G)⩽⌈log2ψ(G)⌉+2. It is not clear whether the upper bound of ⌈log2ψ(G)⌉+2 is tight: till now we are unable to find any interval graph with cub(G)&gt;⌈log2ψ(G)⌉. We also show that for an interval graph G, cub(G)⩽⌈log2α⌉, where α is the independence number of G. Therefore, in the special case of ψ(G)=α, cub(G) is exactly ⌈log2α2⌉. The concept of cubicity can be generalized by considering boxes instead of cubes. A b‐dimensional box is a Cartesian product I1×I2×···×Ib, where each Ii is a closed interval on the real line. The boxicity of a graph, denoted box(G), is the minimum k such that G is the intersection graph of k‐dimensional boxes. It is clear that box(G)⩽cub(G). From the above result, it follows that for any graph G, cub(G)⩽box(G)⌈log2α⌉. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 323–333, 2010

Boolean Graph Operator B

Let G = ( V , E ) be a simple graph with vertex set V and edge set E. Let D be a subset of V, and let B ( D ) be the set of neighbours of D in V ∖ D . The differential ∂ ( D ) of D is defined as | B ( D ) | − | D | . The maximum value of ∂ ( D ) taken over all subsets D ⊆ V is the differential ∂ ( G ) of G. For β ∈ ( − 1 , Δ ) , the β-differential ∂ β ( G ) of G is the maximum value of { | B ( D ) | − β | D | : D ⊆ V } . Motivated by an influential maximization problem, in this paper we study the β -differential of G.