18,436 research outputs found

    The Hamiltonian index of a graph and its branch-bonds

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    Let GG be an undirected and loopless finite graph that is not a path. The minimum mm such that the iterated line graph Lm(G)L^m(G) is hamiltonian is called the hamiltonian index of G,G, denoted by h(G).h(G). A reduction method to determine the hamiltonian index of a graph GG with h(G)2h(G)\geq 2 is given here. With it we will establish a sharp lower bound and a sharp upper bound for h(G)h(G), 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 h(G)h(G) may reach all integers between the lower bound and the upper bound. \u

    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 s0s \ge 0 and t0t \ge 0, a graph GG is (s,t)(s,t)-supereulerian if for any disjoint edge sets X,YE(G)X, Y \subseteq E(G) with Xs|X|\le s and Yt|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 t3t \ge 3 in terms of edge-connectivities, and show that when t3t \ge 3, (s,t)(s,t)-supereulerianicity is polynomially determinable. In Chapter 3, for a subset YE(G)Y \subseteq E(G) with Yκ2˘7(G)1|Y|\le \kappa\u27(G)-1, a necessary and sufficient condition for GYG-Y to be a contractible configuration for supereulerianicity is obtained. We also characterize the (s,t)(s,t)-supereulerianicity of GG when s+tκ2˘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+tκ2˘7(G)s+t\le \kappa\u27(G). For a non-negative integer sV(G)3s\le |V(G)|-3, a graph GG is ss-Hamiltonian if the removal of any ksk\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 r4r\ge 4 even, the minimum degree δk2\delta\ge k\ge 2 and kr+2k\neq r+2, if λ2(H)(r1)δr2(k1)n4(r+1)(nr1)\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

    The Salesman's Improved Tours for Fundamental Classes

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    Finding the exact integrality gap α\alpha for the LP relaxation of the metric Travelling Salesman Problem (TSP) has been an open problem for over thirty years, with little progress made. It is known that 4/3α3/24/3 \leq \alpha \leq 3/2, and a famous conjecture states α=4/3\alpha = 4/3. For this problem, essentially two "fundamental" classes of instances have been proposed. This fundamental property means that in order to show that the integrality gap is at most ρ\rho for all instances of metric TSP, it is sufficient to show it only for the instances in the fundamental class. However, despite the importance and the simplicity of such classes, no apparent effort has been deployed for improving the integrality gap bounds for them. In this paper we take a natural first step in this endeavour, and consider the 1/21/2-integer points of one such class. We successfully improve the upper bound for the integrality gap from 3/23/2 to 10/710/7 for a superclass of these points, as well as prove a lower bound of 4/34/3 for the superclass. Our methods involve innovative applications of tools from combinatorial optimization which have the potential to be more broadly applied

    Upper bounds on the k-forcing number of a graph

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    Given a simple undirected graph GG and a positive integer kk, the kk-forcing number of GG, denoted Fk(G)F_k(G), is the minimum number of vertices that need to be initially colored so that all vertices eventually become colored during the discrete dynamical process described by the following rule. Starting from an initial set of colored vertices and stopping when all vertices are colored: if a colored vertex has at most kk non-colored neighbors, then each of its non-colored neighbors becomes colored. When k=1k=1, this is equivalent to the zero forcing number, usually denoted with Z(G)Z(G), a recently introduced invariant that gives an upper bound on the maximum nullity of a graph. In this paper, we give several upper bounds on the kk-forcing number. Notable among these, we show that if GG is a graph with order n2n \ge 2 and maximum degree Δk\Delta \ge k, then Fk(G)(Δk+1)nΔk+1+min{δ,k}F_k(G) \le \frac{(\Delta-k+1)n}{\Delta - k + 1 +\min{\{\delta,k\}}}. This simplifies to, for the zero forcing number case of k=1k=1, Z(G)=F1(G)ΔnΔ+1Z(G)=F_1(G) \le \frac{\Delta n}{\Delta+1}. Moreover, when Δ2\Delta \ge 2 and the graph is kk-connected, we prove that Fk(G)(Δ2)n+2Δ+k2F_k(G) \leq \frac{(\Delta-2)n+2}{\Delta+k-2}, which is an improvement when k2k\leq 2, and specializes to, for the zero forcing number case, Z(G)=F1(G)(Δ2)n+2Δ1Z(G)= F_1(G) \le \frac{(\Delta -2)n+2}{\Delta -1}. These results resolve a problem posed by Meyer about regular bipartite circulant graphs. Finally, we present a relationship between the kk-forcing number and the connected kk-domination number. As a corollary, we find that the sum of the zero forcing number and connected domination number is at most the order for connected graphs.Comment: 15 pages, 0 figure

    Simpler, faster and shorter labels for distances in graphs

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    We consider how to assign labels to any undirected graph with n nodes such that, given the labels of two nodes and no other information regarding the graph, it is possible to determine the distance between the two nodes. The challenge in such a distance labeling scheme is primarily to minimize the maximum label lenght and secondarily to minimize the time needed to answer distance queries (decoding). Previous schemes have offered different trade-offs between label lengths and query time. This paper presents a simple algorithm with shorter labels and shorter query time than any previous solution, thereby improving the state-of-the-art with respect to both label length and query time in one single algorithm. Our solution addresses several open problems concerning label length and decoding time and is the first improvement of label length for more than three decades. More specifically, we present a distance labeling scheme with label size (log 3)/2 + o(n) (logarithms are in base 2) and O(1) decoding time. This outperforms all existing results with respect to both size and decoding time, including Winkler's (Combinatorica 1983) decade-old result, which uses labels of size (log 3)n and O(n/log n) decoding time, and Gavoille et al. (SODA'01), which uses labels of size 11n + o(n) and O(loglog n) decoding time. In addition, our algorithm is simpler than the previous ones. In the case of integral edge weights of size at most W, we present almost matching upper and lower bounds for label sizes. For r-additive approximation schemes, where distances can be off by an additive constant r, we give both upper and lower bounds. In particular, we present an upper bound for 1-additive approximation schemes which, in the unweighted case, has the same size (ignoring second order terms) as an adjacency scheme: n/2. We also give results for bipartite graphs and for exact and 1-additive distance oracles

    Simulating sparse Hamiltonians with star decompositions

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    We present an efficient algorithm for simulating the time evolution due to a sparse Hamiltonian. In terms of the maximum degree d and dimension N of the space on which the Hamiltonian H acts for time t, this algorithm uses (d^2(d+log* N)||Ht||)^{1+o(1)} queries. This improves the complexity of the sparse Hamiltonian simulation algorithm of Berry, Ahokas, Cleve, and Sanders, which scales like (d^4(log* N)||Ht||)^{1+o(1)}. To achieve this, we decompose a general sparse Hamiltonian into a small sum of Hamiltonians whose graphs of non-zero entries have the property that every connected component is a star, and efficiently simulate each of these pieces.Comment: 11 pages. v2: minor correction
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