67 research outputs found

    Acyclicity in edge-colored graphs

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    A walk WW in edge-colored graphs is called properly colored (PC) if every pair of consecutive edges in WW 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 ii is a proper superset of graphs of acyclicity of type i+1i+1, i=1,2,3,4.i=1,2,3,4. 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

    Master index of volumes 161–170

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    r-Simple k-Path and Related Problems Parameterized by k/r

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    Subject Index Volumes 1–200

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    On Generalizations of Supereulerian Graphs

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    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 κ2˘7(G)\kappa\u27(G) and δ(G)\delta(G) be the edge-connectivity and the minimum degree of a graph GG, respectively. For integers s≥0s \ge 0 and t≥0t \ge 0, a graph GG is (s,t)(s,t)-supereulerian if for any disjoint edge sets X,Y⊆E(G)X, Y \subseteq E(G) with ∣X∣≤s|X|\le s and ∣Y∣≤t|Y|\le t, GG has a spanning closed trail that contains XX and avoids YY. This dissertation is devoted to providing some results on (s,t)(s,t)-supereulerian graphs and supereulerian hypergraphs. In Chapter 2, we determine the value of the smallest integer j(s,t)j(s,t) such that every j(s,t)j(s,t)-edge-connected graph is (s,t)(s,t)-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 (s,t)(s,t)-supereulerian graphs when t≥3t \ge 3 in terms of edge-connectivities, and show that when t≥3t \ge 3, (s,t)(s,t)-supereulerianicity is polynomially determinable. In Chapter 3, for a subset Y⊆E(G)Y \subseteq E(G) with ∣Y∣≤κ2˘7(G)−1|Y|\le \kappa\u27(G)-1, a necessary and sufficient condition for G−YG-Y to be a contractible configuration for supereulerianicity is obtained. We also characterize the (s,t)(s,t)-supereulerianicity of GG when s+t≤κ2˘7(G)s+t\le \kappa\u27(G). These results are applied to show that if GG is (s,t)(s,t)-supereulerian with κ2˘7(G)=δ(G)≥3\kappa\u27(G)=\delta(G)\ge 3, then for any permutation α\alpha on the vertex set V(G)V(G), the permutation graph α(G)\alpha(G) is (s,t)(s,t)-supereulerian if and only if s+t≤κ2˘7(G)s+t\le \kappa\u27(G). For a non-negative integer s≤∣V(G)∣−3s\le |V(G)|-3, a graph GG is ss-Hamiltonian if the removal of any k≤sk\le s vertices results in a Hamiltonian graph. Let is,t(G)i_{s,t}(G) and hs(G)h_s(G) denote the smallest integer ii such that the iterated line graph Li(G)L^{i}(G) is (s,t)(s,t)-supereulerian and ss-Hamiltonian, respectively. In Chapter 4, for a simple graph GG, we establish upper bounds for is,t(G)i_{s,t}(G) and hs(G)h_s(G). Specifically, the upper bound for the ss-Hamiltonian index hs(G)h_s(G) 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 GG is Hamiltonian if and only if GG 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 HH defined by Rodr\\u27iguez in 2002, let λ2(H)\lambda_2(H) be the second largest adjacency eigenvalue of HH. In Chapter 6, we prove that for an integer kk and a rr-uniform hypergraph HH of order nn with r≥4r\ge 4 even, the minimum degree δ≥k≥2\delta\ge k\ge 2 and k≠r+2k\neq r+2, if λ2(H)≤(r−1)δ−r2(k−1)n4(r+1)(n−r−1)\lambda_2(H)\le (r-1)\delta-\frac{r^2(k-1)n}{4(r+1)(n-r-1)}, then HH is kk-edge-connected. %κ2˘7(H)≥k\kappa\u27(H)\ge k. 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 (s,t)(s,t)-supereulerianicity of hypergraphs is another interesting topic to be investigated in the future
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