Showing papers on "Petersen graph published in 2011"
1 Jan 2011
TL;DR: In this paper, the authors showed that the generalized Petersen graph is 4-ordered and the Heawood graph is k-ordered-4-ordered in the sense that the required cycle is a Hamiltonian cycle in the graph.
Abstract: a b s t r a c t A graph G is k-ordered if for every sequence of k distinct vertices of G, there exists a cycle in G containing these k vertices in the specified order. It is k-ordered-Hamiltonian if, in addition, the required cycle is a Hamiltonian cycle in G. The question of the existence of an infinite class of 3-regular 4-ordered-Hamiltonian graphs was posed in Ng and Schultz in 1997 (2). At the time, the only known examples of such graphs were K4 and K3,3. Some progress was made by Meszaros in 2008 (21) when the Petersen graph was found to be 4-ordered and the Heawood graph was proved to be 4-ordered-Hamiltonian; moreover, an infinite class of 3-regular 4-ordered graphs was found. In 2010, a subclass of the generalized Petersen graphs was shown to be 4-ordered in Hsu et al. (9), with an infinite subset of this subclass being 4-ordered-Hamiltonian, thus answering the open question. However, these graphs are bipartite. In this paper we extend the result to another subclass of the generalized Petersen graphs. In particular, we find the first class of infinite non-bipartite graphs that are both 4-ordered-Hamiltonian and 4-ordered-Hamiltonian-connected, which can be seen as a solution to an extension of the question posted in Ng and Schultz in 1997 (2). (A graph G is k-ordered-Hamiltonian-connected if for every sequence of k distinct vertices a1, a2, . . . , ak of G, there exists a Hamiltonian path in G from a1 to ak where these k vertices appear in the specified order.)
49 citations
TL;DR: Upper bounds for the power domination number of some families of Cartesian products of graphs are found: the cylinders Pn□Cm for integers n ≥ 2, m ≥ 3, and the tori Cn⩽CmFor integers n,m ≥ 3 and it is proved those upper bounds provide the exact values of thePower domination numbers if the integers m,n, and k satisfy some given relations.
Abstract: A set S of vertices is defined to be a power dominating set (PDS) of a graph G if every vertex and every edge in G can be monitored by the set S according to a set of rules for power system monitoring. The minimum cardinality of a PDS of G is its power domination number. In this article, we find upper bounds for the power domination number of some families of Cartesian products of graphs: the cylinders Pn□Cm for integers n ≥ 2, m ≥ 3, and the tori Cn□Cm for integers n,m ≥ 3. We apply similar techniques to present upper bounds for the power domination number of generalized Petersen graphs P(m,k). We prove those upper bounds provide the exact values of the power domination numbers if the integers m,n, and k satisfy some given relations. © 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 58(1), 43–49 2011 © 2011 Wiley Periodicals, Inc.
49 citations
TL;DR: It is shown that the only 3-connected internally 4-connected graph of girth 5 in which every odd cycle of length greater than 5 has a chord is the Petersen graph, and Robertson's conjecture is proved.
Abstract: Robertson has conjectured that the only 3-connected internally 4-connected graph of girth 5 in which every odd cycle of length greater than 5 has a chord is the Petersen graph. We prove this conjecture in the special case where the graphs involved are also cubic. Moreover, this proof does not require the internal-4-connectivity assumption. An example is then presented to show that the assumption of internal 4-connectivity cannot be dropped as an hypothesis in the original conjecture. We then summarize our results aimed toward the solution of the conjecture in its original form. In particular, let $G$ be any 3-connected internally-4-connected graph of girth 5 in which every odd cycle of length greater than 5 has a chord. If $C$ is any girth cycle in $G$ then $N(C)\backslash V(C)$ cannot be edgeless, and if $N(C) \backslash V(C)$ contains a path of length at least 2, then the conjecture is true. Consequently, if the conjecture is false and $H$ is a counterexample, then for any girth cycle $C$ in $H$, $N(C) \backslash V(C)$ induces a nontrivial matching $M$ together with an independent set of vertices. Moreover, $M$ can be partitioned into (at most) two disjoint non-empty sets where we can precisely describe how these sets are attached to cycle $C$.
29 citations
1 Jan 2011
TL;DR: In this article, the authors extend the definition and some results given in [1] to a more general total torsion element graph case, where the basic properties and possible structures of the graph T((M)) are studied.
Abstract: The total graph of a commutative ring have been introduced and studied by D. F. Anderson and A. Badawi in [1]. In a manner analogous to a commutative ring, the total torsion element graph of a module M over a commtative ring R can be defined as the undirected graph T(( M)). The basic properties and possible structures of the graph T(( M)) are studied. The main purpose of this paper is to extend the definition and some results given in [1] to a more general total torsion element graph case.
27 citations
Journal Article•
TL;DR: The domination polynomials of cubic graphs of order 10 have been studied in this paper, and it is shown that the Petersen graph is determined uniquely by its domination number of dominating sets.
Abstract: Let G be a simple graph of order n. The domination polynomial of G is the polynomial D(G,x)=\sumi=g(G)n d(G,i) xi, where d(G,i) is the number of dominating sets of G of size i, and g(G) is the domination number of G. In this paper we study the domination polynomials of cubic graphs of order 10. As a consequence, we show that the Petersen graph is determined uniquely by its domination polynomial.
26 citations
TL;DR: It is proved that the maximum number of cliques in an n -vertex graph embeddable in Σ is between 8 ( n - ω) + 2 ω and 8 n + 5 2 2π� + o (2 ω ) , where ω is the maximum integer such that the complete graph K ω embeds inΣ.
Abstract: This paper studies the following question: given a surface Σ and an integer n , what is the maximum number of cliques in an n -vertex graph embeddable in Σ ? We characterise the extremal graphs for this question, and prove that the answer is between 8 ( n - ω ) + 2 ω and 8 n + 5 2 2 ω + o ( 2 ω ) , where ω is the maximum integer such that the complete graph K ω embeds in Σ For the surfaces S 0 , S 1 , S 2 , N 1 , N 2 , N 3 and N 4 we establish an exact answer
24 citations
1 Jan 2011
TL;DR: In this paper, the atom-bond connectivity index and the sum-connectivity index of trees are defined as ABC = ∑ uv∈E √ (du + dv − 2)/(du dv) and χ = ∆ uv ∈E 1/ √ du+dv, respectively.
Abstract: Let G = (V,E) be a graph, du the degree of its vertex u , and uv the edge connecting the vertices u and v . The atom–bond connectivity index and the sum–connectivity index of G are defined as ABC = ∑ uv∈E √ (du + dv − 2)/(du dv) and χ = ∑ uv∈E 1/ √ du + dv , respectively. Continuing the recent researches on ABC [B. Furtula, A. Graovac, D. Vukicevic, Atom-bond connectivity index of trees, Discr. Appl. Math. 157 (2009) 2828–2835] and χ [B. Zhou, N. Trinajstic, On a novel connectivity index, J. Math. Chem. 46 (2009) 1252–1270] we obtain novel upper bounds on these vertex–degree–based graph invariants.
23 citations
TL;DR: Robiano, E.A. Martins, and I. Gutman as mentioned in this paper generalized Fiedler's lemma to graph spectra and graph energy, and applied it to graph energy.
Abstract: In a previous paper [M. Robbiano, E.A. Martins, and I. Gutman, Extending a theorem by Fiedler and applications to graph energy, MATCH Commun. Math. Comput. Chem. 64 (2010), pp. 145–156], a lemma by Fiedler was used to obtain eigenspaces of graphs, and applied to graph energy. In this article Fiedler's lemma is generalized and this generalization is applied to graph spectra and graph energy.
17 citations
TL;DR: In this paper, the authors obtained certain fundamental properties of the total graph on ℤ n and the independent number and clique number of T Γ (Ω n ).
Abstract: Let R be a commutative ring and Z(R) be its set of zero-divisors. The total graph of R, denoted by T Γ (R), is the (undirected) graph with vertices R, and for distinct x, y ∈ R, the vertices x and y are adjacent if and only if x + y ∈ Z(R). In this paper we obtain certain fundamental properties of the total graph on ℤ n . Also we find independent number and clique number of T Γ (ℤ n ).
16 citations
TL;DR: In this paper, the radius of the total graph of a commutative ring R in the case when this graph is connected is determined, and the relation between the radius and the length of the graph is analyzed.
Abstract: We discuss the determination of the radius of the total graph of a
commutative ring R in the case when this graph is connected. Typical
extensions such as polynomial rings, formal power series, idealization of the
R-module M and relations between the total graph of the ring R and its
extensions are also dealt with.
14 citations
TL;DR: It is shown that the Petersen graph is the only Petersen power which can be embedded into the projective plane.
Abstract: For each k, 1≤k≤n, we construct a dot product of n copies of the Petersen graph whose orientable genus is precisely k. We show that these are all possible values for the genus of Pn. This result gives counterexamples of all possible genera to a conjecture of Tinsley and Watkins from 1982. We show that the Petersen graph is the only Petersen power which can be embedded into the projective plane. For each k, 2≤k≤n − 1, we construct a Petersen power Pn whose non-orientable genus is precisely k. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:1-8, 2011
TL;DR: It is shown that the prism over Ok is hamiltonian for all k even and not just for n = 2k + 1, where n is a special case of Kneser graph.
Abstract: The Kneser graph K(n, k) has as its vertex set all k-subsets of an n-set and two k-subsets are adjacent if they are disjoint. The odd graph Ok is a special case of Kneser graph when n = 2k + 1. A long standing conjecture claims that Ok is hamiltonian for all k>2. We show that the prism over Ok is hamiltonian for all k even. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:177-188, 2011 © 2011 Wiley Periodicals, Inc.
TL;DR: It is proved that any double-critical $8$-chromatic graph contains a minor isomorphic to $K_8$ with at most one edge missing, and it is observed that any two-critical graph with minimum degree different from $10 and $11$ contains a $K-8$ minor.
Abstract: A connected $k$-chromatic graph $G$ is said to be double-critical if for all edges $uv$ of $G$ the graph $G - u - v$ is $(k-2)$-colourable. A longstanding conjecture of Erdős and Lovasz states that the complete graphs are the only double-critical graphs. Kawarabayashi, Pedersen and Toft [ Electron. J. Combin. , 17(1): Research Paper 87, 2010] proved that every double-critical $k$-chromatic graph with $k \leq 7$ contains a $K_k$ minor. It remains unknown whether an arbitrary double-critical $8$-chromatic graph contains a $K_8$ minor, but in this paper we prove that any double-critical $8$-chromatic contains a minor isomorphic to $K_8$ with at most one edge missing. In addition, we observe that any double-critical $8$-chromatic graph with minimum degree different from $10$ and $11$ contains a $K_8$ minor.
TL;DR: This paper proves that conjecture is true for n=5,6 and that bubble-sort graph B"n is a union of n-22 edge-disjoint hamiltonian cycles and its perfect matching that has no edges in common with the hamiltonia cycles.
TL;DR: In this article, an inaccessible, vertex transitive, locally finite graph is described, which is not quasi-isometric to a Cayley graph, and it is not an inaccessible graph.
Abstract: An inaccessible, vertex transitive, locally finite graph is described. This graph is not quasi-isometric to a Cayley graph.
TL;DR: Three new upper bounds on the chromatic number are introduced, based on the number of edges and nodes, and is to be applied to any connected component of the graph, whereas @z and @h are based onThe degree of the nodes in the graph.
TL;DR: In this paper, it was shown that every 2-connected projective-planar cubic graph has a closed 2-cell embedding in some orientable surface, which implies both the Cycle Double Cover Conjecture and the Strong Embedding CONjecture.
Abstract: In a closed2-cell embedding of a graph each face is homeomorphic to an open disk and is bounded by a cycle in the graph. The Orientable Strong Embedding Conjecture says that every 2-connected graph has a closed 2-cell embedding in some orientable surface. This implies both the Cycle Double Cover Conjecture and the Strong Embedding Conjecture. In this paper we prove that every 2-connected projective-planar cubic graph has a closed 2-cell embedding in some orientable surface. The three main ingredients of the proof are (1) a surgical method to convert nonorientable embeddings into orientable embeddings; (2) a reduction for 4-cycles for orientable closed 2-cell embeddings, or orientable cycle double covers, of cubic graphs; and (3) a structural result for projective-planar embeddings of cubic graphs. We deduce that every 2-edge-connected projective-planar graph (not necessarily cubic) has an orientable cycle double cover.
Posted Content•
8 Nov 2011
TL;DR: In this paper, it was shown that in each bridgeless cubic graph there exist five perfect matchings covering a portion of the edges at least equal to 215, which is the smallest known upper bound.
Abstract: Let G be a bridgeless cubic graph. A well-known conjecture of Berge and Fulkerson can be stated as follows: there exist five perfect matchings of G such that each edge of G is contained in at least one of them. Here, we prove that in each bridgeless cubic graph there exist five perfect matchings covering a portion of the edges at least equal to 215 . By a generalization of this result, we decrease the best known upper bound, expressed in terms of the size of the graph, for the number of perfect matchings needed to cover the edge-set of G.
Journal Article•
TL;DR: This work considers the case in which G is a cubic graph, and gives conditions for determining P1F graphs within a subfamily of generalized Petersen graphs.
Abstract: A perfectly one-factorable (P1F) regular graph G is a graph admitting a partition of the edge-set into one-factors such that the union of any two of them is a Hamiltonian cycle. We consider the case in which G is a cubic graph. The existence of a P1F cubic graph is guaranteed for each admissible value of the number of vertices. We give conditions for determining P1F graphs within a subfamily of generalized Petersen graphs.
TL;DR: It is shown that if G is not Hamiltonian, then G is either the Petersen graph or contains a 2-factor with a cycle of length at least 7, and infinite families of 2- and 3-connected cubic graphs in which every 2-Factor consists of cycles of length 10 and 16 are given.
Abstract: Every 2-connected cubic graph G has a 2-factor, and much effort has gone into studying conditions that guarantee G to be Hamiltonian. We show that if G is not Hamiltonian, then G is either the Petersen graph or contains a 2-factor with a cycle of length at least 7. We also give infinite families of, respectively, 2- and 3-connected cubic graphs in which every 2-factor consists of cycles of length at most, respectively, 10 and 16.
1 Feb 2011
TL;DR: In this paper, it was shown that if every connected T-vertex-transitive cubic symmetric graph is T-normal, then every connected G-verticestransitive Cayley graph is G-normal.
Abstract: Let Γ be a graph and let G be a vertex-transitive subgroup of the full automorphism group Aut(Γ) of Γ. The graph Γ is called G-normal if G is normal in Aut(Γ). In particular, a Cayley graph Cay(G, S) on a group G with respect to S is normal if the Cayley graph is R(G)-normal, where R(G) is the right regular representation of G. Let T be a non-abelian simple group and let G = Tl with l ≥ 1. We prove that if every connected T-vertex-transitive cubic symmetric graph is T-normal, then every connected G-vertex-transitive cubic symmetric graph is G-normal. This result, among others, implies that a connected cubic symmetric Cayley graph on G is normal except for T ≅ A47 and a connected cubic G-symmetric graph is G-normal except for T ≅ A7, A15 or PSL(4, 2).
TL;DR: The exact k-distance paired domination number of generalized Petersen graphs P(n,1) and P( n,2) for all k≥1 is determined.
Abstract: Let G=(V,E) be a graph without an isolated vertex. A set D?V(G) is a k -distance paired dominating set of G if D is a k-distance dominating set of G and the induced subgraph ?D? has a perfect matching. The minimum cardinality of a k-distance paired dominating set for graph G is the k -distance paired domination number, denoted by ? p k (G). In this paper, we determine the exact k-distance paired domination number of generalized Petersen graphs P(n,1) and P(n,2) for all k?1.
Journal Article•
TL;DR: In this article, the upper bound of the adjacent vertex strongly distinguishing total chromatic number of the graph is obtained for a given graph, and the same upper bound can be obtained for any graph.
23 Mar 2011
TL;DR: Here, one of the new architecture Super Strongly Perfect Graph (SSP) is analyzed and some of its family members namely, Wheel graph, Double Wheelgraph, Cycles, Triangulated graphs and Circulant graphs are classified.
Abstract: Network is the engineering discipline concerned with the communication between computer systems or devices. A computer network is any set of computers or devices connected to each other with the ability to exchange data. Computers on a network are sometimes called nodes. Networks can be broadly classified as using graphs. In the most common sense of the term, a graph is an ordered pair G = (V, E) comprising a set V of vertices or nodes together with a set E of edges or lines, which are 2-element subsets of V. In graph theory, we have many architectures namely Complete graph, Regular graph, Petersen graph, Trees etc. Here we have analyzed one of the new architecture Super Strongly Perfect Graph (SSP). By investigating, we have classified some of its family members namely, Wheel graph, Double Wheel graph, Cycles, Triangulated graphs and Circulant graphs.
27 Jun 2011
TL;DR: This work studies several properties concerning plane topological subgraphs of T n, a complete topological graph with n vertices v1,v2,…,v n in which two edges v i v j (i < j) and v s v t (s < t) cross each other if and only if i < s < t < j or s < i < j < t.
Abstract: The twisted graph T n is a complete topological graph with n vertices v1,v2,…,v n in which two edges v i v j (i < j) and v s v t (s < t) cross each other if and only if i < s < t < j or s < i < j < t. We study several properties concerning plane topological subgraphs of T n .
TL;DR: In this paper, it was shown that the chromatic number of a graph is at least as hard as graph automorphism, but no harder than graph isomorphism when k = 2.
Journal Article•
Posted Content•
TL;DR: In this paper, a local study of domination in graphs was initiated, where the domination value of a vertex is defined to be the number of dominating sets to which the vertex belongs.
Abstract: A set $D \subseteq V(G)$ is a \emph{dominating set} of $G$ if every vertex not in $D$ is adjacent to at least one vertex in $D$. A dominating set of $G$ of minimum cardinality is called a $\gamma(G)$-set. For each vertex $v \in V(G)$, we define the \emph{domination value} of $v$ to be the number of $\gamma(G)$-sets to which $v$ belongs. In this paper, we study some basic properties of the domination value function, thus initiating \emph{a local study of domination} in graphs. Further, we characterize domination value for the Petersen graph, complete $n$-partite graphs, cycles, and paths.
TL;DR: In this article, the authors introduced and investigated the total graph of an ǫ -module over a commutative ring and extended the definition and results given in (Anderson and Badawi, 2008) to more generalize the case of a complete lattice.
Abstract: Let 𝐿 be a complete lattice. We introduce and investigate
the 𝐿 -total graph of an 𝐿 -module over an 𝐿 -commutative ring. The main
purpose of this paper is to extend the definition and results given in (Anderson and Badawi, 2008) to
more generalize the 𝐿 -total graph of an 𝐿 -module case.