56 research outputs found

    Extremal critically connected matroids

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    AbstractA connected matroid M is called a critically connected matroid if the deletion of any one element from M results in a disconnected matroid. We show that a critically connected matroid of rank n, n≥3, can have at most 2n−2 elements. We also show that a critically connected matroid of rank n on 2n−2 elements is isomorphic to the forest matroid of K2, n−2

    Some extremal connectivity results for matroids

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    Let n be an integer exceeding one and M be a matroid having at least n + 2 elements. In this paper, we prove that every n-element subset X of E(M) is in an (n + 1)-element circuit if and only if (i) for every such subset, M X is disconnected, and (ii) for every subset Y with at most n elements, M Y is connected. Various extensions and consequences of this result are also derived including characterizations in terms of connectivity of the 4-point line and of Murty\u27s Sylvester matroids. The former is a result of Seymour. © 1991

    On some extremal connectivity results for graphs and matroids

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    Dirac and Halin have shown for n = 2 and n = 3 respectively that a minimally n-connected graph G has at least ((n-1)|V(G)|-2n)/(2n-1) vertices of degree n. This paper determines the graphs which are extremal with respect to these two results and, in addition, establishes a similar extremal result for minimally connected matroids. © 1982

    Master index of volumes 61–70

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    The Properties of Graphs of Matroids

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    Graph Theory

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    Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures

    On packing minors into connected matroids

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    Let N be a matroid with k connected components and M be a minor-minimal connected matroid having N as a minor. This note proves that |E(M) - E(N)| is at most 2k - 2 unless N or its dual is free, in which case |E(M) - E(N)| ≤k - 1. Examples are given to show that these bounds are best possible for all choices for N. A consequence of the main result is that a minimally connected matroid of rank r and maximum circuit size c has at most 2r - c + 2 elements. This bound sharpens a result of Murty. © 1998 Elsevier Science B.V. All rights reserved

    Zero-free regions for multivariate Tutte polynomials (alias Potts-model partition functions) of graphs and matroids

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    The chromatic polynomial P_G(q) of a loopless graph G is known to be nonzero (with explicitly known sign) on the intervals (-\infty,0), (0,1) and (1,32/27]. Analogous theorems hold for the flow polynomial of bridgeless graphs and for the characteristic polynomial of loopless matroids. Here we exhibit all these results as special cases of more general theorems on real zero-free regions of the multivariate Tutte polynomial Z_G(q,v). The proofs are quite simple, and employ deletion-contraction together with parallel and series reduction. In particular, they shed light on the origin of the curious number 32/27.Comment: LaTeX2e, 49 pages, includes 5 Postscript figure
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