24,724 research outputs found
On the s-Hamiltonian index of a graph
In modeling communication networks by graphs, the problem of designing s-fault-tolerant networks becomes the search for s-Hamiltonian graphs. This thesis is a study of the s-Hamiltonian index of a graph G.;A path P of G is called an arc in G if all the internal vertices of P are divalent vertices of G. We define l (G) = max{lcub}m : G has an arc of length m that is not both of length 2 and in a K3{rcub}. We show that if a connected graph G is not a path, a cycle or K1,3, then for a given s, we give the best known bound of the s-Hamiltonian index of the graph
On Eulerian subgraphs and hamiltonian line graphs
A graph {\color{black}} is Hamilton-connected if for any pair of distinct vertices {\color{black}}, {\color{black}} has a spanning -path; {\color{black}} is 1-hamiltonian if for any vertex subset with , has a spanning cycle. Let , and denote the minimum degree, the matching number and the line graph of a graph , respectively. The following result is obtained. {\color{black} Let be a simple graph} with . If , then each of the following holds. \\ (i) is Hamilton-connected if and only if . \\ (ii) is 1-hamiltonian if and only if . %==========sp For a graph , an integer and distinct vertices , an -path-system of is a subgraph consisting of internally disjoint -paths. The spanning connectivity is the largest integer such that for any with and for any with , has a spanning -path-system. It is known that , and determining if is an NP-complete problem. A graph is maximally spanning connected if . Let and be the smallest integers and such that is maximally spanning connected and , respectively. We show that every locally-connected line graph with connectivity at least 3 is maximally spanning connected, and that the spanning connectivity of a locally-connected line graph can be polynomially determined. As applications, we also determined best possible upper bounds for and , and characterized the extremal graphs reaching the upper bounds. %==============st For integers and , a graph is -supereulerian if for any disjoint edge sets with and , has a spanning closed trail that contains and avoids . Pulleyblank in [J. Graph Theory, 3 (1979) 309-310] showed that determining whether a graph is -supereulerian, even when restricted to planar graphs, is NP-complete. Settling an open problem of Bauer, Catlin in [J. Graph Theory, 12 (1988) 29-45] showed that every simple graph on vertices with , when is sufficiently large, is -supereulerian or is contractible to . We prove the following for any nonnegative integers and . \\ (i) For any real numbers and with , there exists a family of finitely many graphs \F(a,b;s,t) such that if is a simple graph on vertices with and , then either is -supereulerian, or is contractible to a member in \F(a,b;s,t). \\ (ii) Let denote the connected loopless graph with two vertices and parallel edges. If is a simple graph on vertices with and , then when is sufficiently large, either is -supereulerian, or for some integer with , is contractible to a . %==================index For a hamiltonian property \cp, Clark and Wormold introduced the problem of investigating the value \cp(a,b) = \max\{\min\{n: L^n(G) has property \cp\}: and , and proposed a few problems to determine \cp(a,b) with when \cp is being hamiltonian, edge-hamiltonian and hamiltonian-connected. Zhan in 1986 proved that the line graph of a 4-edge-connected graph is Hamilton-connected, which implies a solution to the unsettled cases of above-mentioned problem. We consider an extended version of the problem. Let denote the essential edge-connectivity of a graph , and define \cp\u27(a,b) = \max\{\min\{n: L^n(G) has property \cp\}: and . We investigate the values of \cp\u27(a,b) when \cp is one of these hamiltonian properties. In particular, we show that for any values of , \cp\u27(4,b) \le 2 and \cp\u27(4,b) = 1 if and only if Thomassen\u27s conjecture that every 4-connected line graph is hamiltonian is valid
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
Hamilton decompositions of 6-regular abelian Cayley graphs
In 1969, Lovasz asked whether every connected, vertex-transitive graph has a Hamilton path. This question has generated a considerable amount of interest, yet remains vastly open. To date, there exist no known connected, vertex-transitive graph that does not possess a Hamilton path. For the Cayley graphs, a subclass of vertex-transitive graphs, the following conjecture was made:
Weak Lovász Conjecture: Every nontrivial, finite, connected Cayley graph is hamiltonian.
The Chen-Quimpo Theorem proves that Cayley graphs on abelian groups flourish with Hamilton cycles, thus prompting Alspach to make the following conjecture:
Alspach Conjecture: Every 2k-regular, connected Cayley graph on a finite abelian group has a Hamilton decomposition.
Alspach’s conjecture is true for k = 1 and 2, but even the case k = 3 is still open. It is this case that this thesis addresses.
Chapters 1–3 give introductory material and past work on the conjecture. Chapter 3 investigates the relationship between 6-regular Cayley graphs and associated quotient graphs. A proof of Alspach’s conjecture is given for the odd order case when k = 3. Chapter 4 provides a proof of the conjecture for even order graphs with 3-element connection sets that have an element generating a subgroup of index 2, and having a linear dependency among the other generators.
Chapter 5 shows that if Γ = Cay(A, {s1, s2, s3}) is a connected, 6-regular, abelian Cayley graph of even order, and for some1 ≤ i ≤ 3, Δi = Cay(A/(si), {sj1 , sj2}) is 4-regular, and Δi ≄ Cay(ℤ3, {1, 1}), then Γ has a Hamilton decomposition. Alternatively stated, if Γ = Cay(A, S) is a connected, 6-regular, abelian Cayley graph of even order, then Γ has a Hamilton decomposition if S has no involutions, and for some s ∈ S, Cay(A/(s), S) is 4-regular, and of order at least 4.
Finally, the Appendices give computational data resulting from C and MAGMA programs used to generate Hamilton decompositions of certain non-isomorphic Cayley graphs on low order abelian groups
The Hamiltonian index of a graph and its branch-bonds
Let be an undirected and loopless finite graph that is not a path. The minimum such that the iterated line graph is hamiltonian is called the hamiltonian index of denoted by A reduction method to determine the hamiltonian index of a graph with is given here. With it we will establish a sharp lower bound and a sharp upper bound for , respectively, which improves some known results of P.A. Catlin et al. [J. Graph Theory 14 (1990)] and H.-J. Lai [Discrete Mathematics 69 (1988)]. Examples show that may reach all integers between the lower bound and the upper bound. \u
Algorithm and Complexity for a Network Assortativity Measure
We show that finding a graph realization with the minimum Randi\'c index for
a given degree sequence is solvable in polynomial time by formulating the
problem as a minimum weight perfect b-matching problem. However, the
realization found via this reduction is not guaranteed to be connected.
Approximating the minimum weight b-matching problem subject to a connectivity
constraint is shown to be NP-Hard. For instances in which the optimal solution
to the minimum Randi\'c index problem is not connected, we describe a heuristic
to connect the graph using pairwise edge exchanges that preserves the degree
sequence. In our computational experiments, the heuristic performs well and the
Randi\'c index of the realization after our heuristic is within 3% of the
unconstrained optimal value on average. Although we focus on minimizing the
Randi\'c index, our results extend to maximizing the Randi\'c index as well.
Applications of the Randi\'c index to synchronization of neuronal networks
controlling respiration in mammals and to normalizing cortical thickness
networks in diagnosing individuals with dementia are provided.Comment: Added additional section on application
- …