118 research outputs found

    Contractions, Removals and How to Certify 3-Connectivity in Linear Time

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    It is well-known as an existence result that every 3-connected graph G=(V,E) on more than 4 vertices admits a sequence of contractions and a sequence of removal operations to K_4 such that every intermediate graph is 3-connected. We show that both sequences can be computed in optimal time, improving the previously best known running times of O(|V|^2) to O(|V|+|E|). This settles also the open question of finding a linear time 3-connectivity test that is certifying and extends to a certifying 3-edge-connectivity test in the same time. The certificates used are easy to verify in time O(|E|).Comment: preliminary versio

    Minimal Connectivity

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    A k-connected graph such that deleting any edge / deleting any vertex / contracting any edge results in a graph which is not k-connected is called minimally / critically / contraction-critically k-connected. These three classes play a prominent role in graph connectivity theory, and we give a brief introduction with a light emphasis on reduction- and construction theorems for classes of k-connected graphs.Comment: IMADA-preprint-math, 33 page

    Transitivity is not a (big) restriction on homotopy types

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    For every simplicial complex K there exists a vertex-transitive simplicial complex homotopy equivalent to a wedge of copies of K with some copies of the circle. It follows that every simplicial complex can occur as a homotopy wedge summand in some vertex-transitive complex. One can even demand that the vertex-transitive complex is the clique complex of a Cayley graph or that it is facet-transitive

    On Hardness of the Joint Crossing Number

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    The Joint Crossing Number problem asks for a simultaneous embedding of two disjoint graphs into one surface such that the number of edge crossings (between the two graphs) is minimized. It was introduced by Negami in 2001 in connection with diagonal flips in triangulations of surfaces, and subsequently investigated in a general form for small-genus surfaces. We prove that all of the commonly considered variants of this problem are NP-hard already in the orientable surface of genus 6, by a reduction from a special variant of the anchored crossing number problem of Cabello and Mohar

    On Eulerian subgraphs and hamiltonian line graphs

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    A graph {\color{black}GG} is Hamilton-connected if for any pair of distinct vertices {\color{black}u,v∈V(G)u, v \in V(G)}, {\color{black}GG} has a spanning (u,v)(u,v)-path; {\color{black}GG} is 1-hamiltonian if for any vertex subset S⊆V(G)S \subseteq {\color{black}V(G)} with ∣S∣≤1|S| \le 1, G−SG - S has a spanning cycle. Let δ(G)\delta(G), α2˘7(G)\alpha\u27(G) and L(G)L(G) denote the minimum degree, the matching number and the line graph of a graph GG, respectively. The following result is obtained. {\color{black} Let GG be a simple graph} with ∣E(G)∣≥3|E(G)| \ge 3. If δ(G)≥α2˘7(G)\delta(G) \geq \alpha\u27(G), then each of the following holds. \\ (i) L(G)L(G) is Hamilton-connected if and only if κ(L(G))≥3\kappa(L(G))\ge 3. \\ (ii) L(G)L(G) is 1-hamiltonian if and only if κ(L(G))≥3\kappa(L(G))\ge 3. %==========sp For a graph GG, an integer s≥0s \ge 0 and distinct vertices u,v∈V(G)u, v \in V(G), an (s;u,v)(s; u, v)-path-system of GG is a subgraph HH consisting of ss internally disjoint (u,v)(u,v)-paths. The spanning connectivity κ∗(G)\kappa^*(G) is the largest integer ss such that for any kk with 0≤k≤s0 \le k \le s and for any u,v∈V(G)u, v \in V(G) with u≠vu \neq v, GG has a spanning (k;u,v)(k; u,v)-path-system. It is known that κ∗(G)≤κ(G)\kappa^*(G) \le \kappa(G), and determining if κ∗(G)3˘e0\kappa^*(G) \u3e 0 is an NP-complete problem. A graph GG is maximally spanning connected if κ∗(G)=κ(G)\kappa^*(G) = \kappa(G). Let msc(G)msc(G) and sk(G)s_k(G) be the smallest integers mm and m2˘7m\u27 such that Lm(G)L^m(G) is maximally spanning connected and κ∗(Lm2˘7(G))≥k\kappa^*(L^{m\u27}(G)) \ge k, respectively. We show that every locally-connected line graph with connectivity at least 3 is maximally spanning connected, and that the spanning connectivity of a locally-connected line graph can be polynomially determined. As applications, we also determined best possible upper bounds for msc(G)msc(G) and sk(G)s_k(G), and characterized the extremal graphs reaching the upper bounds. %==============st 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. Pulleyblank in [J. Graph Theory, 3 (1979) 309-310] showed that determining whether a graph is (0,0)(0,0)-supereulerian, even when restricted to planar graphs, is NP-complete. Settling an open problem of Bauer, Catlin in [J. Graph Theory, 12 (1988) 29-45] showed that every simple graph GG on nn vertices with δ(G)≥n5−1\delta(G) \ge \frac{n}{5} -1, when nn is sufficiently large, is (0,0)(0,0)-supereulerian or is contractible to K2,3K_{2,3}. We prove the following for any nonnegative integers ss and tt. \\ (i) For any real numbers aa and bb with 03˘ca3˘c10 \u3c a \u3c 1, there exists a family of finitely many graphs \F(a,b;s,t) such that if GG is a simple graph on nn vertices with κ2˘7(G)≥t+2\kappa\u27(G) \ge t+2 and δ(G)≥an+b\delta(G) \ge an + b, then either GG is (s,t)(s,t)-supereulerian, or GG is contractible to a member in \F(a,b;s,t). \\ (ii) Let ℓK2\ell K_2 denote the connected loopless graph with two vertices and ℓ\ell parallel edges. If GG is a simple graph on nn vertices with κ2˘7(G)≥t+2\kappa\u27(G) \ge t+2 and δ(G)≥n2−1\delta(G) \ge \frac{n}{2}-1, then when nn is sufficiently large, either GG is (s,t)(s,t)-supereulerian, or for some integer jj with t+2≤j≤s+tt+2 \le j \le s+t, GG is contractible to a jK2j K_2. %==================index For a hamiltonian property \cp, Clark and Wormold introduced the problem of investigating the value \cp(a,b) = \max\{\min\{n: L^n(G) has property \cp\}: κ2˘7(G)≥a\kappa\u27(G) \ge a and δ(G)≥b}\delta(G) \ge b\}, and proposed a few problems to determine \cp(a,b) with b≥a≥4b \ge a \ge 4 when \cp is being hamiltonian, edge-hamiltonian and hamiltonian-connected. Zhan in 1986 proved that the line graph of a 4-edge-connected graph is Hamilton-connected, which implies a solution to the unsettled cases of above-mentioned problem. We consider an extended version of the problem. Let ess2˘7(G)ess\u27(G) denote the essential edge-connectivity of a graph GG, and define \cp\u27(a,b) = \max\{\min\{n: L^n(G) has property \cp\}: ess2˘7(G)≥aess\u27(G) \ge a and δ(G)≥b}\delta(G) \ge b\}. We investigate the values of \cp\u27(a,b) when \cp is one of these hamiltonian properties. In particular, we show that for any values of b≥1b \ge 1, \cp\u27(4,b) \le 2 and \cp\u27(4,b) = 1 if and only if Thomassen\u27s conjecture that every 4-connected line graph is hamiltonian is valid

    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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