scholarly journals Circuit-Difference Matroids

10.37236/9314 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
George Drummond ◽  
Tara Fife ◽  
Kevin Grace ◽  
James Oxley
Keyword(s):  
Disjoint Union ◽  
Symmetric Difference ◽  
Binary Matroids ◽  
Regular Matroid

One characterization of binary matroids is that the symmetric difference of every pair of intersecting circuits is a disjoint union of circuits. This paper considers circuit-difference matroids, that is, those matroids in which the symmetric difference of every pair of intersecting circuits is a single circuit. Our main result shows that a connected regular matroid is circuit-difference if and only if it contains no pair of skew circuits. Using a result of Pfeil, this enables us to explicitly determine all regular circuit-difference matroids. The class of circuit-difference matroids is not closed under minors, but it is closed under series minors. We characterize the infinitely many excluded series minors for the class.

A characterization of n-connected splitting matroids

2014 ◽  
Vol 07 (04) ◽  
pp. 1450060
Author(s):  
P. P. Malavadkar ◽  
M. M. Shikare ◽  
S. B. Dhotre
Keyword(s):  
Binary Matroid ◽  
Binary Matroids ◽  
Splitting Operation

The splitting operation on an n-connected binary matroid may not yield an n-connected binary matroid. In this paper, we characterize n-connected binary matroids which yield n-connected binary matroids by the generalized splitting operation.

scholarly journals A characterization of the existence of a Souslin line

1982 ◽  
Vol 25 (3) ◽  
pp. 425-431
Author(s):  
Nobuyuki Kemoto
Keyword(s):  
Disjoint Union ◽  
Chain Condition ◽  
First Category ◽  
Countable Chain Condition ◽  
Isolated Points ◽  
Countable Chain

The main purpose of this paper is to show that there exists a Souslin line if and only if there exists a countable chain condition space which is not weak-separable but has a generic π-base. If I is the closure of the isolated points in a space X, then X is said to be weak-separable if a first category set is dense in X – I. A π-base is said to be generic if, whenever a member of is included in the disjoint union of members of it is included in one of them.

scholarly journals Semiarcs with Long Secants

10.37236/3771 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Bence Csajbók
Keyword(s):  
Lower Bound ◽  
Projective Plane ◽  
Complete Characterization ◽  
Symmetric Difference ◽  
Blocking Set ◽  
Point Set

In a projective plane $\Pi_q$ of order $q$, a non-empty point set $\mathcal{S}_t$ is a $t$-semiarc if the number of tangent lines to $\mathcal{S}_t$ at each of its points is $t$. If $\mathcal{S}_t$ is a $t$-semiarc in $\Pi_q$, $t<q$, then each line intersects $\mathcal{S}_t$ in at most $q+1-t$ points. Dover proved that semiovals (semiarcs with $t=1$) containing $q$ collinear points exist in $\Pi_q$ only if $q\leq 3$. We show that if $t>1$, then $t$-semiarcs with $q+1-t$ collinear points exist only if $t\geq \sqrt{q-1}$. In $\mathrm{PG}(2,q)$ we prove the lower bound $t\geq(q-1)/2$, with equality only if $\mathcal{S}_t$ is a blocking set of Rédei type of size $3(q+1)/2$.We call the symmetric difference of two lines, with $t$ further points removed from each line, a $V_t$-configuration. We give conditions ensuring a $t$-semiarc to contain a $V_t$-configuration and give the complete characterization of such $t$-semiarcs in $\mathrm{PG}(2,q)$.

scholarly journals Semigroups with atomistic congruence lattices

1992 ◽  
Vol 52 (1) ◽  
pp. 88-102 ◽  
Author(s):  
Karl Auinger
Keyword(s):  
Congruence Lattice ◽  
Disjoint Union ◽  
Partition Lattice ◽  
Congruence Lattices

AbstractIn this note a characterization of semigroups with atomistic consruence lattices, given for weakly reductive semigroups, is generalized to arbitrary semigroups. Also, it is shown that there is a complete congruence on the congruence lattice of such a semigroup that decomposes it into a disjoint union of intervals of the partition lattice.

scholarly journals Idealness of k-wise intersecting families

2020 ◽  
Author(s):  
Ahmad Abdi ◽  
Gérard Cornuéjols ◽  
Tony Huynh ◽  
Dabeen Lee
Keyword(s):  
Chromatic Number ◽  
Common Element ◽  
Binary Matroids ◽  
Projective Geometries ◽  
Intersecting Families ◽  
Cycle Covers ◽  
Binary Clutters

Abstract A clutter is k-wise intersecting if every k members have a common element, yet no element belongs to all members. We conjecture that, for some integer $$k\ge 4$$ k ≥ 4 , every k-wise intersecting clutter is non-ideal. As evidence for our conjecture, we prove it for $$k=4$$ k = 4 for the class of binary clutters. Two key ingredients for our proof are Jaeger’s 8-flow theorem for graphs, and Seymour’s characterization of the binary matroids with the sums of circuits property. As further evidence for our conjecture, we also note that it follows from an unpublished conjecture of Seymour from 1975. We also discuss connections to the chromatic number of a clutter, projective geometries over the two-element field, uniform cycle covers in graphs, and quarter-integral packings of value two in ideal clutters.

scholarly journals Properties of $\theta$-super positive graphs

10.37236/2041 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Cheng Yeaw Ku ◽  
Kok Bin Wong
Keyword(s):  
Disjoint Union ◽  
Perfect Matching ◽  
Matching Theory ◽  
Matching Polynomial

Let the matching polynomial of a graph $G$ be denoted by $\mu (G,x)$. A graph $G$ is said to be $\theta$-super positive if  $\mu(G,\theta)\neq 0$ and $\mu(G\setminus v,\theta)=0$ for all $v\in V(G)$. In particular, $G$ is $0$-super positive if and only if $G$ has a perfect matching. While much is known about $0$-super positive graphs, almost nothing is known about $\theta$-super positive graphs for $\theta \neq 0$. This motivates us to investigate the structure of $\theta$-super positive graphs in this paper. Though a $0$-super positive graph need not contain any cycle, we show that a $\theta$-super positive graph with $\theta \neq 0$ must contain a cycle. We introduce two important types of $\theta$-super positive graphs, namely $\theta$-elementary and $\theta$-base graphs. One of our main results is that any $\theta$-super positive graph $G$ can be constructed by adding certain type of edges to a disjoint union of $\theta$-base graphs; moreover, these $\theta$-base graphs are uniquely determined by $G$. We also give a characterization of $\theta$-elementary graphs: a graph $G$ is $\theta$-elementary if and only if the set of all its $\theta$-barrier sets form a partition of $V(G)$. Here, $\theta$-elementary graphs and $\theta$-barrier sets can be regarded as $\theta$-analogue of elementary graphs and Tutte sets in classical matching theory.

scholarly journals K-Regular Matroids

2021 ◽  
Author(s):  
◽  
Charles A Semple
Keyword(s):  
Finite Number ◽  
Prime Power ◽  
Negative Integer ◽  
Regular Matroid ◽  
Excluded Minor ◽  
Excluded Minors

<p>The class of matroids representable over all fields is the class of regular matroids. The class of matroids representable over all fields except perhaps GF(2) is the class of near-regular matroids. Let k be a non-negative integer. This thesis considers the class of k-regular matroids, a generalization of the last two classes. Indeed, the classes of regular and near-regular matroids coincide with the classes of 0-regular and 1-regular matroids, respectively. This thesis extends many results for regular and near-regular matroids. In particular, for all k, the class of k-regular matroids is precisely the class of matroids representable over a particular partial field. Every 3-connected member of the classes of either regular or near-regular matroids has a unique representability property. This thesis extends this property to the 3-connected members of the class of k-regular matroids for all k. A matroid is [omega] -regular if it is k-regular for some k. It is shown that, for all k [greater than or equal to] 0, every 3-connected k-regular matroid is uniquely representable over the partial field canonically associated with the class of [omega] -regular matroids. To prove this result, the excluded-minor characterization of the class of k-regular matroids within the class of [omega] -regular matroids is first proved. It turns out that, for all k, there are a finite number of [omega] -regular excluded minors for the class of k-regular matroids. The proofs of the last two results on k-regular matroids are closely related. The result referred to next is quite different in this regard. The thesis determines, for all r and all k, the maximum number of points that a simple rank-r k-regular matroid can have and identifies all such matroids having this number. This last result generalizes the corresponding results for regular and near-regular matroids. Some of the main results for k-regular matroids are obtained via a matroid operation that is a generalization of the operation of [Delta] - Y exchange. This operation is called segment-cosegment exchange and, like the operation of [Delta] - Y exchange, has a dual operation. This thesis defines the generalized operation and its dual, and identifies many of their attractive properties. One property in particular, is that, for a partial field P, the set of excluded minors for representability over P is closed under the operations of segment-cosegment exchange and its dual. This result generalizes the corresponding result for [Delta] - Y and Y - [Delta] exchanges. Moreover, a consequence of it is that, for a prime power q, the number of excluded minors for GF(q)-representability is at least 2q-4.</p>

scholarly journals ℓ-matrices and a characterization of binary matroids

1974 ◽  
Vol 8 (2) ◽  
pp. 139-145 ◽  
Author(s):  
Robert E. Bixby
Keyword(s):  
Binary Matroids
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