Product Of Complete Graphs Research Articles

Relating the Outer-Independent Total Roman Domination Number with Some Classical Parameters of Graphs

For a given graph G without isolated vertex we consider a function f: V(G) rightarrow {0,1,2}. For every iin {0,1,2}, let V_i={vin V(G):; f(v)=i}. The function f is known to be an outer-independent total Roman dominating function for the graph G if it is satisfied that; (i) every vertex in V_0 is adjacent to at least one vertex in V_2; (ii) V_0 is an independent set; and (iii) the subgraph induced by V_1cup V_2 has no isolated vertex. The minimum possible weight omega (f)=sum _{vin V(G)}f(v) among all outer-independent total Roman dominating functions for G is called the outer-independent total Roman domination number of G. In this article we obtain new tight bounds for this parameter that improve some well-known results. Such bounds can also be seen as relationships between this parameter and several other classical parameters in graph theory like the domination, total domination, Roman domination, independence, and vertex cover numbers. In addition, we compute the outer-independent total Roman domination number of Sierpiński graphs, circulant graphs, and the Cartesian and direct products of complete graphs.

Cartesian Products of Some Regular Graphs Admitting Antimagic Labeling for Arbitrary Sets of Real Numbers

An edge labeling of graph G with labels in A is an injection from E G to A , where E G is the edge set of G , and A is a subset of ℝ . A graph G is called ℝ -antimagic if for each subset A of ℝ with A = E G , there is an edge labeling with labels in A such that the sums of the labels assigned to edges incident to distinct vertices are different. The main result of this paper is that the Cartesian products of complete graphs (except K 1 ) and cycles are ℝ -antimagic.

Estimates for the Norm of Generalized Maximal Operator on Strong Product of Graphs

LetG=G1×G2×⋯×Gmbe the strong product of simple, finite connected graphs, and letϕ:ℕ⟶0,∞be an increasing function. We consider the action of generalized maximal operatorMGϕonℓpspaces. We determine the exact value ofℓp-quasi-norm ofMGϕfor the case whenGis strong product of complete graphs, where0<p≤1. However, lower and upper bounds ofℓp-norm have been determined when1<p<∞. Finally, we computed the lower and upper bounds ofMGϕpwhenGis strong product of arbitrary graphs, where0<p≤1.

On star coloring of modular product of graphs

A star coloring of a graph $G$ is a proper vertex coloring in which every path on four vertices in $G$ is not bicolored. The star chromatic number $\chi_{s}\left(G\right)$ of $G$ is the least number of colors needed to star color $G$. In this paper, we find the exact values of the star chromatic number of modular product of complete graph with complete graph $K_m \diamond K_n$, path with complete graph $P_m \diamond K_n$ and star graph with complete graph $K_{1,m}\diamond K_n$. \par All graphs in this paper are finite, simple, connected and undirected graph and we follow \cite{bm, cla, f} for terminology and notation that are not defined here. We denote the vertex set and the edge set of $G$ by $V(G)$ and $E(G)$, respectively. Branko Gr\"{u}nbaum introduced the concept of star chromatic number in 1973. A star coloring \cite{alberton, fertin, bg} of a graph $G$ is a proper vertex coloring in which every path on four vertices uses at least three distinct colors. The star chromatic number $\chi_{s}\left(G\right)$ of $G$ is the least number of colors needed to star color $G$. \par During the years star coloring of graphs has been studied extensively by several authors, for instance see \cite{alberton, col, fertin}.

Decomposition of product graphs into paths and stars on five vertices

Let Sk and Kk respectively denote a path, a star and a complete graph on k vertices. By a -decomposition of a graph G, we mean a decomposition of G into r copies of and s copies of In this paper, it shown that the graph admits a -decomposition if and only if where denotes a tensor product of complete graphs. Also we extend the existence of such a decomposition in complete m-partite graphs.

Proper Coloring Distance in Edge-Colored Cartesian Products of Complete Graphs and Cycles

An path that is edge-colored is called proper if no two consecutive edges receive the same color. A general graph that is edge-colored is called properly connected if, for every pair of vertices in the graph, there exists a properly colored path from one to the other. Given two vertices u and v in a properly connected graph G, the proper distance is the length of the shortest properly colored path from u to v. By considering a specific class of colorings that are properly connected for Cartesian products of complete and cyclic graphs, we present results on the proper distance between all pairs of vertices in the graph.

A Note on the Steinerk-Diameter of Tensor Product Networks

Given a graph [Formula: see text] and [Formula: see text], the Steiner distance [Formula: see text] is the minimum size among all connected subgraphs of [Formula: see text] whose vertex sets contain [Formula: see text]. The Steiner [Formula: see text]-diameter [Formula: see text] is the maximum value of [Formula: see text] among all sets of [Formula: see text] vertices. In this short note, we study the Steiner [Formula: see text]-diameters of the tensor product of complete graphs.

Slightly subcritical hypercube percolation

We study bond percolation on the hypercube {0,1}m in the slightly subcritical regime where p = pc(1 − εm) and εm = o(1) but εm ≫ 2−m/3 and study the clusters of largest volume and diameter. We establish that with high probability the largest component has cardinality , that the maximal diameter of all clusters is , and that the maximal mixing time of all clusters is .These results hold in different levels of generality, and in particular, some of the estimates hold for various classes of graphs such as high‐dimensional tori, expanders of high degree and girth, products of complete graphs, and infinite lattices in high dimensions.

完全图的笛卡尔积的广义3-连通度

完全图的笛卡尔积的广义3-连通度

P3-factorization of triangulated Cartesian product of complete graphs

Partition of G into edge-disjoint H-factors is called an H-factorization of G. Shehzad Afzal and Clemens Brand have factorized graphs into factors which are isomorphic to triangulated Cartesian product of two subgraphs. Also they have discussed properties of triangulated Cartesian product of more than two graphs. In this paper, we show that the necessary conditions mn ≡ 0 (mod 3), m, n are odd and 3(mn + m + n - 3) ≡ 0 (mod 8) are sufficient for the existence of a P3-factorization of Km ⧅ Kn, where ⧅ denotes triangulated Cartesian product of graphs, if one of the following holds: (i) m ≡ 9 (mod 12), n = 5, 13 and 17, (ii) m ≡ 9 (mod 12), ns, s > 1, n = 5, 13 and 17, (iii) m ≡ 9 (mod 12), n = psqt for all s, t ≥ 1, where p = 5, 13 and q = 13, 17, p ≠ q, (iv) m ≡ 9 (mod 12), n ≡ 9 (mod 12).

Harmonious Coloring on Corona Product of Complete Graphs

In this paper, we find the harmonious chromatic number of the corona product of any graph $$G$$ of order $$l$$ with the complete graph $$K_n$$ for $$l \le n$$ . As a consequence of this work, we also obtain the harmonious chromatic number of $$t$$ copies of $$K_n$$ for $$t \le n+1$$ .

Distance three labellings for K n ×K 2

In a wireless communication network, the channel assignment problem addresses the correct assignment of a frequency to each transmitter in the network. To avoid interference between two nearby transmitters, the assigned frequencies must satisfy certain conditions related to the distance between transmitters. We can use only a limited number of channels; hence, we have to minimize the range of frequencies used. The distance labelling of a graph is a mathematical model of the channel assignment problem derived from the work of Hale [Frequency assignment: Theory and application, Proc. IEEE 68 (1980), pp. 1497–1514]. Lam et al. [L(j, k)-labelings for the products of complete graphs, J. Comb. Optim. 14 (2007), pp. 219–227] addressed distance two labelling for the direct product K n ×K m of complete graphs K n and K m . In this paper, we solve the distance three labelling problem for K n ×K 2.

On the Acyclic Chromatic Number of Hamming Graphs

An acyclic coloring of a graph G is a proper coloring of the vertex set of G such that G contains no bichromatic cycles. The acyclic chromatic number of a graph G is the minimum number k such that G has an acyclic coloring with k colors. In this paper, acyclic colorings of Hamming graphs, products of complete graphs, are considered. Upper and lower bounds on the acyclic chromatic number of Hamming graphs are given.

Distance monotonicity and a new characterization of Hamming graphs

Hamming graphs are simply Cartesian products of complete graphs. Several characterizations of these graphs involve the notion of intervals in a graph and related topics. The notion of interval distance monotonicity has already been used in order to obtain a characterization of hypercubes. Here, we revisit this notion and use it to obtain a new characterization of Hamming graphs by performing a combination of the hypercube characterization and the interval hypercube characterization.

On Hamilton cycle decompositions of the tensor product of complete graphs

In this paper, we show that the tensor product of complete graphs is hamilton cycle decomposable.

Cartesian products of block graphs

Hamming graphs (being the Cartesian products of complete graphs) are known to be the quasi-median graphs not containing the 3-vertex path as a convex subgraph. Similarly, the Cartesian products of trees have been characterized among median graphs by a single forbidden convex subgraph. In this note a common generalization of these two results is given that characterizes the Cartesian products of block graphs.

Skeletons, recognition algorithm and distance matrix of quasi-median graphs

A quasi-median graph can be characterized as a weak retract of a Cartesian product of complete graphs or equivalently as a graph of finite windex. We derive a new characterization of quasi-median graphs which allows us to recognize these graphs in O(n√n log n+m log n) time. It is shown that skeletons of quasi-median graphs are median graphs. For an arbitrary vertex s an s-skeleton of a graph is obtained from G by deleting all edges uv that satisfy d(u, s) = d(v, s). As a by-product of this approach the size of a maximum complete subgraph of a quasi-median graph can be computed within the same time bound. Furthermore, we show that the distance matrix of a quasi-median graph can be computed in O(n 2).

Vertex Transitivity and Super Line Connectedness

A connected graph X is said to be super line connected if every minimum edge cut is the set of edges incident with some vertex. It is proved that a connected Abelian Cayley graph is super line connected unless it is a cycle or the product of complete graphs, one of size 2 and one of size $m\geqq 3$. This result generalizes previous characterizations for circulants by Boesch and Wang and for hypercubes by Boesch, Lee, Wang, and Yang. A simple algebraic characterization for strongly connected Cayley digraphs that are not super arc connected and a structural characterization for the Abelian case are given. In a previous work, one of the authors presented a characterization of the connected vertex-transitive graphs that are not super line connected. The final result of the present paper generalizes this result to directed graphs.

A Dynamic location problem for graphs

We introduce a class of optimization problems, calleddynamic location problems, involving the processing of requests that occur sequentially at the nodes of a graphG. This leads to the definition of a new parameter of graphs, called the window indexWX(G), that measures how large a “window” into the future is needed to solve every instance of the dynamic location problem onG optimally on-line. We completely characterize this parameter:WX(G)≦k if and only ifG is a weak retract of a product of complete graphs of size at mostk. As a byproduct, we obtain two (polynomially recognizable) structural characterizations of such graphs, extending a result of Bandelt.

Interval-regular graphs

An interval-regular graph is a connected graph in which, for any two vertices u and v, the number of neighbours of u on all shortest ( u, v)-paths equals d( u, v). It is proved that in an interval-regular graph the shortest ( u, v)-paths induce a hypercube of dimension d( u, v), for any two vertices u and v. The products of complete graphs are characterized as interval-regular graphs satisfying some extra conditions. The extended odd graphs are introduced as critical example with respect to the results proved.