N2 Vertices Research Articles

Approximating TSP walks in subcubic graphs

We prove that every simple 2-connected subcubic graph on n vertices with n2 vertices of degree 2 has a TSP walk of length at most 5n+n24−1, confirming a conjecture of Dvořák, Král', and Mohar. This bound is best possible; there are infinitely many subcubic and cubic graphs whose minimum TSP walks have lengths 5n+n24−1 and 5n4−2 respectively. We characterize the extremal subcubic examples meeting this bound. We also give a quadratic-time combinatorial algorithm for finding such a TSP walk. In particular, we obtain a 54-approximation algorithm for the graphic TSP on simple cubic graphs, improving on the previously best known approximation ratio of 97.

The Domination number of 2-neighbourhood-corona graphs

Given a simple graph G. Vertex set of G is V, and edge set of G is E. Domination set, denoted by S, that is subset of V such that every vertex in V which is not element of S has distance one to S. The least number of the elements of S is the domination number of the graph G, that is ϒ(G). Let G1 and G2 be a simple graph. G1 has n1 vertices, and has m1 edges. G2 has n2 vertices, and has m2 edges. We defined an operator called neighbourhood corona, denoted by a star ‘*’. Graph G1*G2 is a new graph obtained by making ni copies of second graph and for each i make connecting all vertices in i-th copy of second graph G2 to neighbours of vi = 1, 2, …, n. Furthermore, new graph 2-neighbourhood corona G1*2G2, has n1 copies of G2 and for each i make connecting to all vertices of ith copy of G2 to neighbours of vi, i = 1, 2, …, n. In this research, we determined ϒ(Gi*2G2) where G1 is a complete graph Kn or Cycle Cn, and G2 is K1 or P2. Furthermore, we determined ϒ(G1 *mG2) due to domination number of complete graph Kn and cycle Cn. Since ϒ(Kn) = 1 then ϒ (Kn*mK1) = 1 + m. Furthermore, ϒ(Kn*mP2) = 1 + m. Since then . Furthermore, .

Isolation number of maximal outerplanar graphs

A subset S of vertices in a graph G is called an isolating set if V(G)∖NG[S] is an independent set of G. The isolation number ι(G) is the minimum cardinality of an isolating set of G. Let G be a maximal outerplanar graph of order n with n2 vertices of degree 2. It was previously proved that ι(G)≤n4. In this paper, we improve this bound to be ι(G)≤n+n25whenn2≤n4,n−n23otherwise,and these bounds are best possible.

Further results on Erdős–Faber–Lovász conjecture

In 1972, Erdős–Faber–Lovász (EFL) conjectured that, if H is a linear hypergraph consisting of n edges of cardinality n, then it is possible to color the vertices with n colors so that no two vertices with the same color are in the same edge. In 1978, Deza, Erdös and Frankl had given an equivalent version of the same for graphs: Let G=⋃i=1nAi denote a graph with n complete graphs A1,A2,…,An, each having exactly n vertices and have the property that every pair of complete graphs has at most one common vertex, then the chromatic number of G is n. The clique degree dK(v) of a vertex v in G is given by dK(v)=|{Ai:v∈V(Ai),1≤i≤n}|. In this paper we give a method for assigning colors to the graphs satisfying the hypothesis of the Erdős–Faber–Lovász conjecture and every Ai (1≤i≤n) has atmost n2 vertices of clique degree greater than one using Symmetric latin Squares and clique degrees of the vertices of G.

Bounds on the burning number

Motivated by a graph theoretic process intended to measure the speed of the spread of contagion in a graph, Bonato et al. (Burning a Graph as a Model of Social Contagion, Lecture Notes in Computer Science 8882 (2014) 13-22) define the burning number b(G) of a graph G as the smallest integer k for which there are vertices x1,…,xk such that for every vertex u of G, there is some i∈{1,…,k} with distG(u,xi)≤k−i, and distG(xi,xj)≥j−i for every i,j∈{1,…,k}.For a connected graph G of order n, they prove that b(G)≤2n−1, and conjecture b(G)≤n. We show that b(G)≤3219⋅n1−ϵ+2719ϵ and b(G)≤12n7+3≈1.309n+3 for every connected graph G of order n and every 0<ϵ<1. For a tree T of order n with n2 vertices of degree 2, and n≥3 vertices of degree at least 3, we show b(T)≤(n+n2)+14+12 and b(T)≤n+n≥3. Furthermore, we characterize the binary trees of depth r that have burning number r+1.

The modified box dimension and average weighted receiving time of the weighted hierarchical graph

In this paper we study the weighted hierarchical graph which is generated from bipartite graph with N=n1+n2 vertices, in which the weights of edges have been assigned to different values with certain scale. Firstly, we introduce the definition of the modified box dimension. Then for the weighted hierarchical graph we deduce the modified box dimension, dimMB({Gn}n∈N)=−logrN, depending on the weighted factor r and the number N of copies. Secondly, we mainly study their two average weighted receiving times (AWRTs), 〈T〉In and 〈T〉IIn, of the weighted hierarchical graph on random walk. We discuss two cases. In the case of n1n2r≠n2−n1, we deduce both AWRTs grow as a power-law function of the network size |V(Gn)| with the postive exponent, represented by θ=logN(Nn1n2) or θ=logNr=1−dimMB({Gn}n∈N), which means that the bigger the value of the modified box dimension is, the slower the process of receiving information is. In the case of n1n2r=n2−n1, both AWRTs tend to constant ( if N<n1n2), the AWRTs grow with increasing order as logN|V(Gn)|( if N=n1n2), and both AWRTs grow as a power-law function of the network size |V(Gn)| with the exponent, represented by θ=logN(Nn1n2)>0( if N>n1n2).

Bichain graphs: Geometric model and universal graphs

Bichain graphs form a bipartite analog of split permutation graphs, also known as split graphs of Dilworth number 2. Unlike graphs of Dilworth number 1 that enjoy many nice properties, split permutation graphs are substantially more complex. To better understand the global structure of split permutation graphs, in the present paper we study their bipartite analog. We show that bichain graphs admit a simple geometric representation and have a universal element of quadratic order, i.e. an n-universal bichain graph with n2 vertices. The latter result improves a recent cubic construction of universal split permutation graphs.

The second largest component in the supercritical 2D Hamming graph

AbstractThe two‐dimensional Hamming graph H(2,n) consists of the n2 vertices (i,j), 1 ≤ i,j ≤ n, two vertices being adjacent when they share a common coordinate. We examine random subgraphs of H(2,n) in percolation with edge probability p, in such a way that the average degree satisfies 2(n − 1)p = 1 + ε. Previous work [8] has shown that in the barely supercritical region n−2/3 ln1/3n ≪ ε ≪ 1, the largest component satisfies a law of large numbers with mean 2εn. Here we show that the second largest component has, with high probability, size bounded by 28ε−2 log(n2ε3), so that the dominant component has emerged. This result also suggests that a discrete duality principle holds, where, after removing the largest connected component in the supercritical regime, the remaining random subgraphs behave as in the subcritical regime. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010

Topological Book Embedding of Bipartite Graphs

A topological book embedding of a graph is an embedding in a book that carries the vertices in the spine of the book and the edges in the pages so that edges are allowed to cross the spine. Recently, the author has shown that for an arbitrary graph G with n vertices there exists a d+1-page book embedding of G in which each edge crosses the spine logdn times. This paper improves the result for the case of bipartite graphs and shows that there exists a d+1-page book embedding of a bipartite graph Gn1,n2 having two partite sets with n1 and n2 vertices respectively n1 ≥ n2 in which each edge crosses the spine [logdn2]-1 times.

The Chirality of Icosahedral Fullerenes: a Comparison of the Tripling (leapfrog), Quadrupling (chamfering), and Septupling (capra) Transformations

Fullerene polyhedra of icosahedral symmetry have the midpoints of their 12 pentagonal faces at the vertices of a macroicosahedron and can be characterized by the patterns of their hexagonal faces on the (triangular) macrofaces of this macroicosahedron. The numbers of the vertices in fullerene polyhedra of icosahedral symmetry satisfy the Goldberg equation v=20(h2+hk+k2), where h and k are two integers and 0 <h≥ k≥ 0 and define a two-dimensional Goldberg vector G = (h, k). The known tripling (leapfrog), quadrupling (chamfering), and septupling (capra) transformations correspond to the Goldberg vectors (1, 1), (2, 0), and (2, 1), respectively. The tripling and quadrupling transformations applied to the regular dodecahedron generate achiral fullerene polyhedra with the full Ih point group. However, the septupling transformation destroys the reflection operations of the underlying icosahedron to generate chiral fullerene polyhedra having only the I icosahedral rotational point group. Generalization of the quadrupling transformation leads to the fundamental homologous series of achiral fullerene polyhedra having 20 n2 vertices and Goldberg vectors (n, 0). A related homologous series of likewise achiral fullerene polyhedra having 60 n2 vertices and Goldberg vectors (n, n) is obtained by applying the tripling transformation to regular dodecahedral C20 to give truncated icosahedral C60 followed by the generalized operations (as in the case of quadrupling) for obtaining homologous series of fullerenes. Generalization of the septupling (capra) transformation leads to a homologous series of chiral C20m fullerenes with the I point group and Goldberg vectors G=(h, 1) where m=h2+h+1.

ON THE CLIQUE-WIDTH OF SOME PERFECT GRAPH CLASSES

Graphs of clique–width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k-expressions based on graph operations which use k vertex labels. In this paper we study the clique–width of perfect graph classes. On one hand, we show that every distance–hereditary graph, has clique–width at most 3, and a 3–expression defining it can be obtained in linear time. On the other hand, we show that the classes of unit interval and permutation graphs are not of bounded clique–width. More precisely, we show that for every [Formula: see text] there is a unit interval graph In and a permutation graph Hn having n2 vertices, each of whose clique–width is at least n. These results allow us to see the border within the hierarchy of perfect graphs between classes whose clique–width is bounded and classes whose clique–width is unbounded. Finally we show that every n×n square grid, [Formula: see text], n ≥ 3, has clique–width exactly n+1.

An efficient algorithm for the Knight's tour problem

A knight’s tour is a series of moves made by a knight visiting every square of an n x n chessboard exactly once. The knight’s tour problem is the problem of constructing such a tour, given n. A knight’s tour is called closed if the last square visited is also reachable from the first square by a knight’s move, and open otherwise. Define the knight’s graph for an n x n chessboard to be the graph G = (V,E), where V={(i,j) ( 1 d i,j Gn}, and E={((i,j),(k,Q) ( {Ii-kl,Ijt[} = {1,2}}. That is, there is a vertex for every square of the board and an edge between two vertices exactly when there is a knight’s move from one to the other. Then, more formally, an open knight’s tour is defined to be a Hamiltonian path, and a closed knight’s tour is defined to be a Hamiltonian cycle on a knight’s graph. A knight’s graph has n2 vertices and 4n2 12n + 8 edges. The formal study of the knight’s tour problem is said to have begun with Euler [lo] in 1759, who considered the standard 8 x 8 chessboard. Rouse Ball and Coxeter [l] give an interesting bibliography of the history of the problem from this point. Dudeney [8, 91 contains a description of exactly which rectangular chessboards have knight’s tours; in particular, an n x n chessboard has a closed knight’s tour iff n Z 6 is even, ’ and an open knight’s tour iff n 3 5. It is not clear who first proved this fact, but it appears to be part of the folklore of the subject (see, for example, [2]). There exist several independently conceived linear time (i.e. O(n2)) algorithms for constructing knight’s tours (see, for example, [5, 191). Takemji and Lee [22, 231 recently proposed a neural network solution to the knight’s tour problem, although it appears to be of little use in practice (see [ 17, 181). We will describe in this paper a new, simple, and fast algorithm for constructing knight’s tours on square boards.