9 research outputs found
Small cycles, generalized prisms and Hamiltonian cycles in the Bubble-sort graph
The Bubble-sort graph , is a Cayley graph over the
symmetric group generated by transpositions from the set . It is a bipartite graph containing all even cycles of
length , where . We give an explicit
combinatorial characterization of all its - and -cycles. Based on this
characterization, we define generalized prisms in , and
present a new approach to construct a Hamiltonian cycle based on these
generalized prisms.Comment: 13 pages, 7 figure
On the k-restricted edge-connectivity of matched sum graphs
A matched sum graph M of two graphs and of the same order n is obtained by adding to the union (or sum) of and a set M of n independent edges which join vertices in V () to vertices in V (). When and are isomorphic, M is just a permutation graph. In this work we derive
bounds for the k-restricted edge connectivity λ(k) of matched sum graphs M for 2 ≤ k ≤ 5, and present some sufficient conditions for the optimality of λ(k)(M).Peer Reviewe
Balanced and 1-balanced graph constructions
AbstractThere are several density functions for graphs which have found use in various applications. In this paper, we examine two of them, the first being given by b(G)=|E(G)|/|V(G)|, and the other being given by g(G)=|E(G)|/(|V(G)|−ω(G)), where ω(G) denotes the number of components of G. Graphs for which b(H)≤b(G) for all subgraphs H of G are called balanced graphs, and graphs for which g(H)≤g(G) for all subgraphs H of G are called 1-balanced graphs (also sometimes called strongly balanced or uniformly dense in the literature). Although the functions b and g are very similar, they distinguish classes of graphs sufficiently differently that b(G) is useful in studying random graphs, g(G) has been useful in designing networks with reduced vulnerability to attack and in studying the World Wide Web, and a similar function is useful in the study of rigidity. First we give a new characterization of balanced graphs. Then we introduce a graph construction which generalizes the Cartesian product of graphs to produce what we call a generalized Cartesian product. We show that generalized Cartesian product derived from a tree and 1-balanced graphs are 1-balanced, and we use this to prove that the generalized Cartesian products derived from 1-balanced graphs are 1-balanced
On Generalizations of Supereulerian Graphs
A graph is supereulerian if it has a spanning closed trail. Pulleyblank in 1979 showed that determining whether a graph is supereulerian, even when restricted to planar graphs, is NP-complete. Let and be the edge-connectivity and the minimum degree of a graph , respectively. For integers and , a graph is -supereulerian if for any disjoint edge sets with and , has a spanning closed trail that contains and avoids . This dissertation is devoted to providing some results on -supereulerian graphs and supereulerian hypergraphs.
In Chapter 2, we determine the value of the smallest integer such that every -edge-connected graph is -supereulerian as follows:
j(s,t) = \left\{ \begin{array}{ll} \max\{4, t + 2\} & \mbox{ if $0 \le s \le 1$, or $(s,t) \in \{(2,0), (2,1), (3,0),(4,0)\}$,} \\ 5 & \mbox{ if $(s,t) \in \{(2,2), (3,1)\}$,} \\ s + t + \frac{1 - (-1)^s}{2} & \mbox{ if $s \ge 2$ and $s+t \ge 5$. } \end{array} \right.
As applications, we characterize -supereulerian graphs when in terms of edge-connectivities, and show that when , -supereulerianicity is polynomially determinable.
In Chapter 3, for a subset with , a necessary and sufficient condition for to be a contractible configuration for supereulerianicity is obtained. We also characterize the -supereulerianicity of when . These results are applied to show that if is -supereulerian with , then for any permutation on the vertex set , the permutation graph is -supereulerian if and only if .
For a non-negative integer , a graph is -Hamiltonian if the removal of any vertices results in a Hamiltonian graph. Let and denote the smallest integer such that the iterated line graph is -supereulerian and -Hamiltonian, respectively. In Chapter 4, for a simple graph , we establish upper bounds for and . Specifically, the upper bound for the -Hamiltonian index sharpens the result obtained by Zhang et al. in [Discrete Math., 308 (2008) 4779-4785].
Harary and Nash-Williams in 1968 proved that the line graph of a graph is Hamiltonian if and only if has a dominating closed trail, Jaeger in 1979 showed that every 4-edge-connected graph is supereulerian, and Catlin in 1988 proved that every graph with two edge-disjoint spanning trees is a contractible configuration for supereulerianicity. In Chapter 5, utilizing the notion of partition-connectedness of hypergraphs introduced by Frank, Kir\\u27aly and Kriesell in 2003, we generalize the above-mentioned results of Harary and Nash-Williams, of Jaeger and of Catlin to hypergraphs by characterizing hypergraphs whose line graphs are Hamiltonian, and showing that every 2-partition-connected hypergraph is a contractible configuration for supereulerianicity.
Applying the adjacency matrix of a hypergraph defined by Rodr\\u27iguez in 2002, let be the second largest adjacency eigenvalue of . In Chapter 6, we prove that for an integer and a -uniform hypergraph of order with even, the minimum degree and , if , then is -edge-connected. %.
Some discussions are displayed in the last chapter. We extend the well-known Thomassen Conjecture that every 4-connected line graph is Hamiltonian to hypergraphs. The -supereulerianicity of hypergraphs is another interesting topic to be investigated in the future
Connectivity of Generalized Prisms Over G
The problem of building larger graphs with a given graph as an induced subgraph is one which can arise in various applications and in particular can be important when constructing large communications networks from smaller ones. What one can conclude from this paper is that generalized prisms over G may provide an important such construction because the connectivity of the newly created graph is larger than that of the original (connected) graph, regardless of the permutation used.
For a graph G with vertices labeled 1,2,…, n and a permutation α in Sn, the generalized prisms over G, α(G) (also called a permutation graph), consists of two copies of G, say Gx and Gy, along with the edges (xi, yα(i), for 1≤i≤n. The purpose of this paper is to examine the connectivity of generalized prisms over G. In particular, upper and lower bounds are found. Also, the connectivity and edge-connectivity are determined for generalized prisms over trees, cycles, wheels, n-cubes, complete graphs, and complete bipartite graphs. Finally, the connectivity of the generalized prism over G, α(G), is determined for all α in the automorphism group of G