67 research outputs found
Acyclicity in edge-colored graphs
A walk in edge-colored graphs is called properly colored (PC) if every
pair of consecutive edges in is of different color. We introduce and study
five types of PC acyclicity in edge-colored graphs such that graphs of PC
acyclicity of type is a proper superset of graphs of acyclicity of type
, The first three types are equivalent to the absence of PC
cycles, PC trails, and PC walks, respectively. While graphs of types 1, 2 and 3
can be recognized in polynomial time, the problem of recognizing graphs of type
4 is, somewhat surprisingly, NP-hard even for 2-edge-colored graphs (i.e., when
only two colors are used). The same problem with respect to type 5 is
polynomial-time solvable for all edge-colored graphs. Using the five types, we
investigate the border between intractability and tractability for the problems
of finding the maximum number of internally vertex disjoint PC paths between
two vertices and the minimum number of vertices to meet all PC paths between
two vertices
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
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