522 research outputs found

    Spanning Trees and Spanning Eulerian Subgraphs with Small Degrees. II

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    Let GG be a connected graph with XV(G)X\subseteq V(G) and with the spanning forest FF. Let λ[0,1]\lambda\in [0,1] be a real number and let η:X(λ,)\eta:X\rightarrow (\lambda,\infty) be a real function. In this paper, we show that if for all SXS\subseteq X, ω(GS)vS(η(v)2)+2λ(eG(S)+1)\omega(G\setminus S)\le\sum_{v\in S}\big(\eta(v)-2\big)+2-\lambda(e_G(S)+1), then GG has a spanning tree TT containing FF such that for each vertex vXv\in X, dT(v)η(v)λ+max{0,dF(v)1}d_T(v)\le \lceil\eta(v)-\lambda\rceil+\max\{0,d_F(v)-1\}, where ω(GS)\omega(G\setminus S) denotes the number of components of GSG\setminus S and eG(S)e_G(S) denotes the number of edges of GG with both ends in SS. This is an improvement of several results and the condition is best possible. Next, we also investigate an extension for this result and deduce that every kk-edge-connected graph GG has a spanning subgraph HH containing mm edge-disjoint spanning trees such that for each vertex vv, dH(v)mk(dG(v)2m)+2md_H(v)\le \big\lceil \frac{m}{k}(d_G(v)-2m)\big\rceil+2m, where k2mk\ge 2m; also if GG contains kk edge-disjoint spanning trees, then HH can be found such that for each vertex vv, dH(v)mk(dG(v)m)+md_H(v)\le \big\lceil \frac{m}{k}(d_G(v)-m)\big\rceil+m, where kmk\ge m. Finally, we show that strongly 22-tough graphs, including (3+1/2)(3+1/2)-tough graphs of order at least three, have spanning Eulerian subgraphs whose degrees lie in the set {2,4}\{2,4\}. In addition, we show that every 11-tough graph has spanning closed walk meeting each vertex at most 22 times and prove a long-standing conjecture due to Jackson and Wormald (1990).Comment: 46 pages, Keywords: Spanning tree; spanning Eulerian; spanning closed walk; connected factor; toughness; total exces

    Factors and Connected Factors in Tough Graphs with High Isolated Toughness

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    In this paper, we show that every 11-tough graph with order and isolated toughness at least r+1r+1 has a factor whose degrees are rr, except for at most one vertex with degree r+1r+1. Using this result, we conclude that every 33-tough graph with order and isolated toughness at least r+1r+1 has a connected factor whose degrees lie in the set {r,r+1}\{r,r+1\}, where r3r\ge 3. Also, we show that this factor can be found mm-tree-connected, when GG is a (2m+ϵ)(2m+\epsilon)-tough graph with order and isolated toughness at least r+1r+1, where r(2m1)(2m/ϵ+1)r\ge (2m-1)(2m/\epsilon+1) and ϵ>0\epsilon > 0. Next, we prove that every (m+ϵ)(m+\epsilon)-tough graph of order at least 2m2m with high enough isolated toughness admits an mm-tree-connected factor with maximum degree at most 2m+12m+1. From this result, we derive that every (2+ϵ)(2+\epsilon)-tough graph of order at least three with high enough isolated toughness has a spanning Eulerian subgraph whose degrees lie in the set {2,4}\{2,4\}. In addition, we provide a family of 5/35/3-tough graphs with high enough isolated toughness having no connected even factors with bounded maximum degree

    Fan-Type Conditions for Collapsible Graphs

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    Approximating ATSP by Relaxing Connectivity

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    The standard LP relaxation of the asymmetric traveling salesman problem has been conjectured to have a constant integrality gap in the metric case. We prove this conjecture when restricted to shortest path metrics of node-weighted digraphs. Our arguments are constructive and give a constant factor approximation algorithm for these metrics. We remark that the considered case is more general than the directed analog of the special case of the symmetric traveling salesman problem for which there were recent improvements on Christofides' algorithm. The main idea of our approach is to first consider an easier problem obtained by significantly relaxing the general connectivity requirements into local connectivity conditions. For this relaxed problem, it is quite easy to give an algorithm with a guarantee of 3 on node-weighted shortest path metrics. More surprisingly, we then show that any algorithm (irrespective of the metric) for the relaxed problem can be turned into an algorithm for the asymmetric traveling salesman problem by only losing a small constant factor in the performance guarantee. This leaves open the intriguing task of designing a "good" algorithm for the relaxed problem on general metrics.Comment: 25 pages, 2 figures, fixed some typos in previous versio
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