401 research outputs found

    Further topics in connectivity

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    Continuing the study of connectivity, initiated in §4.1 of the Handbook, we survey here some (sufficient) conditions under which a graph or digraph has a given connectivity or edge-connectivity. First, we describe results concerning maximal (vertex- or edge-) connectivity. Next, we deal with conditions for having (usually lower) bounds for the connectivity parameters. Finally, some other general connectivity measures, such as one instance of the so-called “conditional connectivity,” are considered. For unexplained terminology concerning connectivity, see §4.1.Peer ReviewedPostprint (published version

    Oriented paths in n-chromatic digraphs

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    In this thesis, we try to treat the problem of oriented paths in n-chromatic digraphs. We first treat the case of antidirected paths in 5-chromatic digraphs, where we explain El-Sahili's theorem and provide an elementary and shorter proof of it. We then treat the case of paths with two blocks in n-chromatic digraphs with n greater than 4, where we explain the two different approaches of Addario-Berry et al. and of El-Sahili. We indicate a mistake in Addario-Berry et al.'s proof and provide a correction for it.Comment: 25 pages, Master thesis in Graph Theory at the Lebanese Universit

    Superconnectivity of Networks Modeled by the Strong Product of Graphs

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    Maximal connectivity and superconnectivity in a network are two important features of its reliability. In this paper, using graph terminology, we first give a lower bound for the vertex connectivity of the strong product of two networks and then we prove that the resulting structure is more reliable than its generators. Namely, sufficient conditions for a strong product of two networks to be maximally connected and superconnected are given.Ministerio de Economía y Competitividad MTM2014-60127-

    On the size of maximally non-hamiltonian digraphs

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    A graph is called maximally non-hamiltonian if it is non-hamiltonian, yet for any two non-adjacent vertices there exists a hamiltonian path between them. In this paper, we naturally extend the concept to directed graphs and bound their size from below and above. Our results on the lower bound constitute our main contribution, while the upper bound can be obtained using a result of Lewin, but we give here a different proof. We describe digraphs attaining the upper bound, but whether our lower bound can be improved remains open

    The mincut graph of a graph

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    In this paper we introduce an intersection graph of a graph GG, with vertex set the minimum edge-cuts of GG. We find the minimum cut-set graphs of some well-known families of graphs and show that every graph is a minimum cut-set graph, henceforth called a \emph{mincut graph}. Furthermore, we show that non-isomorphic graphs can have isomorphic mincut graphs and ask the question whether there are sufficient conditions for two graphs to have isomorphic mincut graphs. We introduce the rr-intersection number of a graph GG, the smallest number of elements we need in SS in order to have a family F={S1,S2,Si}F=\{S_1, S_2 \ldots , S_i\} of subsets, such that Si=r|S_i|=r for each subset. Finally we investigate the effect of certain graph operations on the mincut graphs of some families of graphs

    Maximally Edge-Connected Realizations and Kundu's kk-factor Theorem

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    A simple graph GG with edge-connectivity λ(G)\lambda(G) and minimum degree δ(G)\delta(G) is maximally edge connected if λ(G)=δ(G)\lambda(G)=\delta(G). In 1964, given a non-increasing degree sequence π=(d1,,dn)\pi=(d_{1},\ldots,d_{n}), Jack Edmonds showed that there is a realization GG of π\pi that is kk-edge-connected if and only if dnkd_{n}\geq k with i=1ndi2(n1)\sum_{i=1}^{n}d_{i}\geq 2(n-1) when dn=1d_{n}=1. We strengthen Edmonds's result by showing that given a realization G0G_{0} of π\pi if Z0Z_{0} is a spanning subgraph of G0G_{0} with δ(Z0)1\delta(Z_{0})\geq 1 such that E(Z0)n1|E(Z_{0})|\geq n-1 when δ(G0)=1\delta(G_{0})=1, then there is a maximally edge-connected realization of π\pi with G0E(Z0)G_{0}-E(Z_{0}) as a subgraph. Our theorem tells us that there is a maximally edge-connected realization of π\pi that differs from G0G_{0} by at most n1n-1 edges. For δ(G0)2\delta(G_{0})\geq 2, if G0G_{0} has a spanning forest with cc components, then our theorem says there is a maximally edge-connected realization that differs from G0G_{0} by at most ncn-c edges. As an application we combine our work with Kundu's kk-factor Theorem to find maximally edge-connected realizations with a (k1,,kn)(k_{1},\dots,k_{n})-factor for kkik+1k\leq k_{i}\leq k+1 and present a partial result to a conjecture that strengthens the regular case of Kundu's kk-factor theorem.Comment: 13 pages, 1 figur
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