9 research outputs found
On average connectivity of the strong product of graphs
The average connectivity Îș(G) of a graph G is the average, over all pairs of vertices, of the
maximum number of internally disjoint paths connecting these vertices. The connectivity
Îș(G) can be seen as the minimum, over all pairs of vertices, of the maximum number
of internally disjoint paths connecting these vertices. The connectivity and the average
connectivity are upper bounded by the minimum degree ÎŽ(G) and the average degree d(G)
of G, respectively. In this paper the average connectivity of the strong product G1 G2 of two
connected graphs G1 and G2 is studied. A sharp lower bound for this parameter is obtained.
As a consequence, we prove that Îș(G1 G2) = d(G1 G2) if Îș(Gi) = d(Gi), i = 1, 2. Also
we deduce that Îș(G1 G2) = ÎŽ(G1 G2) if Îș(Gi) = ÎŽ(Gi), i = 1, 2.Ministerio de EducaciĂłn y Ciencia MTM2011-28800-C02-02Generalitat de Cataluña 1298 SGR200
On the connectivity and restricted edge-connectivity of 3-arc graphs
A 3âarc of a graph G is a 4-tuple (y, a, b, x) of vertices such that both (y, a, b) and (a, b, x) are paths of length two in G. Let ââG denote the symmetric digraph of a graph G. The 3-arc graph X(G) of a given graph G is defined to have vertices the arcs of ââG. Two vertices (ay), (bx) are adjacent in X(G) if and only if (y, a, b, x) is a 3-arc of G. The purpose of this work is to study the edge-connectivity and restricted edge-connectivity of 3-arc graphs.We prove that the 3-arc graph X(G) of every connected graph G of minimum degree ÎŽ(G) â„ 3 has edge-connectivity λ(X(G)) â„ (ÎŽ(G) â 1)2; and restricted edge- connectivity λ(2)(X(G)) â„ 2(ÎŽ(G) â 1)2 â 2 if Îș(G) â„ 2. We also provide examples showing that all these bounds are sharp.Peer Reviewe
Further topics in connectivity
Continuing the study of connectivity, initiated in §4.1 of the Handbook, we survey here some (sufficient) conditions under which a graph or digraph has a given connectivity or edge-connectivity. First, we describe results concerning maximal (vertex- or edge-) connectivity. Next, we deal with conditions for having (usually lower) bounds for the connectivity parameters. Finally, some other general connectivity measures, such as one instance of the so-called âconditional connectivity,â are considered.
For unexplained terminology concerning connectivity, see §4.1.Peer ReviewedPostprint (published version
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
A bound on connectivity of iterated line graphs
For simple connected graphs that are neither paths nor cycles, we define l(G) = max{m: G has a divalent path of length m that is not both of length 2 and in a K3}, where a divalent path is a path whose internal vertices have degree two in G. Let G be a graph and Ln(G) be its n-th iterated line graph of G. We use (Formula Presented) and Îș(G) for the essential edge connectivity and vertex connectivity of G, respectively. Let G be a simple connected graph that is not a path, a cycle or K1,3, with l(G) = l â„ 1. We prove that (i) for integers (Formula Presented) (ii) for integers (Formula Presented). The bounds are best possible