4,059 research outputs found

    Properties of minimally tt-tough graphs

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    A graph GG is minimally tt-tough if the toughness of GG is tt and the deletion of any edge from GG decreases the toughness. Kriesell conjectured that for every minimally 11-tough graph the minimum degree ÎŽ(G)=2\delta(G)=2. We show that in every minimally 11-tough graph ÎŽ(G)≀n+23\delta(G)\le\frac{n+2}{3}. We also prove that every minimally 11-tough claw-free graph is a cycle. On the other hand, we show that for every t∈Qt \in \mathbb{Q} any graph can be embedded as an induced subgraph into a minimally tt-tough graph

    Spanning Trees and Spanning Eulerian Subgraphs with Small Degrees. II

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    Let GG be a connected graph with X⊆V(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 S⊆XS\subseteq X, ω(G∖S)≀∑v∈S(η(v)−2)+2−λ(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 v∈Xv\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(G∖S)\omega(G\setminus S) denotes the number of components of G∖SG\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 k≄2mk\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 k≄mk\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

    Minimally toughness in special graph classes

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    Let tt be a positive real number. A graph is called tt-tough, if the removal of any cutset SS leaves at most ∣S∣/t|S|/t components. The toughness of a graph is the largest tt for which the graph is tt-tough. A graph is minimally tt-tough, if the toughness of the graph is tt and the deletion of any edge from the graph decreases the toughness. In this paper we investigate the minimum degree and the recognizability of minimally tt-tough graphs in the class of chordal graphs, split graphs, claw-free graphs and 2K22K_2-free graphs

    Conditions for minimally tough graphs

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    Katona, Solt\'esz, and Varga showed that no induced subgraph can be excluded from the class of minimally tough graphs. In this paper, we consider the opposite question, namely which induced subgraphs, if any, must necessarily be present in each minimally tt-tough graph. Katona and Varga showed that for any rational number t∈(1/2,1]t \in (1/2,1], every minimally tt-tough graph contains a hole. We complement this result by showing that for any rational number t>1t>1, every minimally tt-tough graph must contain either a hole or an induced subgraph isomorphic to the kk-sun for some integer k≄3k \ge 3. We also show that for any rational number t>1/2t > 1/2, every minimally tt-tough graph must contain either an induced 44-cycle, an induced 55-cycle, or two independent edges as an induced subgraph

    On the minimum degree of minimally t t -tough, claw-free graphs

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    A graph G G is minimally t t -tough if the toughness of G G is t t and deletion of any edge from G G decreases its toughness. Katona et al. conjectured that the minimum degree of any minimally t t -tough graph is ⌈2t⌉ \lceil 2t\rceil and proved that the minimum degree of minimally 12 \frac{1}2 -tough and 1 1 -tough, claw-free graphs is 1 and 2, respectively. We have show that every minimally 3/2 3/2 -tough, claw-free graph has a vertex of degree of 3 3 . In this paper, we give an upper bound on the minimum degree of minimally tt-tough, claw-free graphs for t≄2 t\geq 2
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