1,042 research outputs found

    A construction for a counterexample to the pseudo 2-factor isomorphic graph conjecture

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    A graph GG admiting a 22-factor is \textit{pseudo 22-factor isomorphic} if the parity of the number of cycles in all its 22-factors is the same. In [M. Abreu, A.A. Diwan, B. Jackson, D. Labbate and J. Sheehan. Pseudo 22-factor isomorphic regular bipartite graphs. Journal of Combinatorial Theory, Series B, 98(2) (2008), 432-444.] some of the authors of this note gave a partial characterisation of pseudo 22-factor isomorphic bipartite cubic graphs and conjectured that K3,3K_{3,3}, the Heawood graph and the Pappus graph are the only essentially 44-edge-connected ones. In [J. Goedgebeur. A counterexample to the pseudo 22-factor isomorphic graph conjecture. Discr. Applied Math., 193 (2015), 57-60.] Jan Goedgebeur computationally found a graph G\mathscr{G} on 3030 vertices which is pseudo 22-factor isomorphic cubic and bipartite, essentially 44-edge-connected and cyclically 66-edge-connected, thus refuting the above conjecture. In this note, we describe how such a graph can be constructed from the Heawood graph and the generalised Petersen graph GP(8,3)GP(8,3), which are the Levi graphs of the Fano 737_3 configuration and the M\"obius-Kantor 838_3 configuration, respectively. Such a description of G\mathscr{G} allows us to understand its automorphism group, which has order 144144, using both a geometrical and a graph theoretical approach simultaneously. Moreover we illustrate the uniqueness of this graph

    Irreducible pseudo 2-factor isomorphic cubic bipartite graphs

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    A bipartite graph is {\em pseudo 2--factor isomorphic} if all its 2--factors have the same parity of number of circuits. In \cite{ADJLS} we proved that the only essentially 4--edge-connected pseudo 2--factor isomorphic cubic bipartite graph of girth 4 is K3,3K_{3,3}, and conjectured \cite[Conjecture 3.6]{ADJLS} that the only essentially 4--edge-connected cubic bipartite graphs are K3,3K_{3,3}, the Heawood graph and the Pappus graph. There exists a characterization of symmetric configurations n3n_3 %{\bf decide notation and how to use it in the rest of the paper} due to Martinetti (1886) in which all symmetric configurations n3n_3 can be obtained from an infinite set of so called {\em irreducible} configurations \cite{VM}. The list of irreducible configurations has been completed by Boben \cite{B} in terms of their {\em irreducible Levi graphs}. In this paper we characterize irreducible pseudo 2--factor isomorphic cubic bipartite graphs proving that the only pseudo 2--factor isomorphic irreducible Levi graphs are the Heawood and Pappus graphs. Moreover, the obtained characterization allows us to partially prove the above Conjecture

    Embedability between right-angled Artin groups

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    In this article we study the right-angled Artin subgroups of a given right-angled Artin group. Starting with a graph \gam, we produce a new graph through a purely combinatorial procedure, and call it the extension graph \gam^e of \gam. We produce a second graph \gam^e_k, the clique graph of \gam^e, by adding extra vertices for each complete subgraph of \gam^e. We prove that each finite induced subgraph Λ\Lambda of \gam^e gives rise to an inclusion A(\Lambda)\to A(\gam). Conversely, we show that if there is an inclusion A(\Lambda)\to A(\gam) then Λ\Lambda is an induced subgraph of \gam^e_k. These results have a number of corollaries. Let P4P_4 denote the path on four vertices and let CnC_n denote the cycle of length nn. We prove that A(P4)A(P_4) embeds in A(\gam) if and only if P4P_4 is an induced subgraph of \gam. We prove that if FF is any finite forest then A(F)A(F) embeds in A(P4)A(P_4). We recover the first author's result on co--contraction of graphs and prove that if \gam has no triangles and A(\gam) contains a copy of A(Cn)A(C_n) for some n≥5n\geq 5, then \gam contains a copy of CmC_m for some 5≤m≤n5\le m\le n. We also recover Kambites' Theorem, which asserts that if A(C4)A(C_4) embeds in A(\gam) then \gam contains an induced square. Finally, we determine precisely when there is an inclusion A(Cm)→A(Cn)A(C_m)\to A(C_n) and show that there is no "universal" two--dimensional right-angled Artin group.Comment: 35 pages. Added an appendix and a proof that the extension graph is quasi-isometric to a tre

    A counterexample to the pseudo 2-factor isomorphic graph conjecture

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    A graph GG is pseudo 2-factor isomorphic if the parity of the number of cycles in a 2-factor is the same for all 2-factors of GG. Abreu et al. conjectured that K3,3K_{3,3}, the Heawood graph and the Pappus graph are the only essentially 4-edge-connected pseudo 2-factor isomorphic cubic bipartite graphs (Abreu et al., Journal of Combinatorial Theory, Series B, 2008, Conjecture 3.6). Using a computer search we show that this conjecture is false by constructing a counterexample with 30 vertices. We also show that this is the only counterexample up to at least 40 vertices. A graph GG is 2-factor hamiltonian if all 2-factors of GG are hamiltonian cycles. Funk et al. conjectured that every 2-factor hamiltonian cubic bipartite graph can be obtained from K3,3K_{3,3} and the Heawood graph by applying repeated star products (Funk et al., Journal of Combinatorial Theory, Series B, 2003, Conjecture 3.2). We verify that this conjecture holds up to at least 40 vertices.Comment: 8 pages, added some extra information in Discrete Applied Mathematics (2015

    Distance-regular graphs

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    This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

    Taut distance-regular graphs and the subconstituent algebra

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    We consider a bipartite distance-regular graph GG with diameter DD at least 4 and valency kk at least 3. We obtain upper and lower bounds for the local eigenvalues of GG in terms of the intersection numbers of GG and the eigenvalues of GG. Fix a vertex of GG and let TT denote the corresponding subconstituent algebra. We give a detailed description of those thin irreducible TT-modules that have endpoint 2 and dimension D−3D-3. In an earlier paper the first author defined what it means for GG to be taut. We obtain three characterizations of the taut condition, each of which involves the local eigenvalues or the thin irreducible TT-modules mentioned above.Comment: 29 page
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