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This paper addresses the problem of synchronizing orthogonal matrices over directed graphs. For synchronized transformations (or matrices), composite transformations over loops equal the identity. We formulate the synchronization problem as…
This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a)…
We prove that for any circle graph $H$ with at least one edge and for any positive integer $k$, there exists an integer $t=t(k,H)$ so that every graph $G$ either has a vertex-minor isomorphic to the disjoint union of $k$ copies of $H$, or…
We study dynamics and bifurcations of 2-dimensional reversible maps having a symmetric saddle fixed point with an asymmetric pair of nontransversal homoclinic orbits (a symmetric nontransversal homoclinic figure-8). We consider…
A family of periodic perturbations of an attracting robust heteroclinic cycle defined on the two-sphere is studied by reducing the analysis to that of a one-parameter family of maps on a circle. The set of zeros of the family forms a…
Lov\'asz has completely characterised the structure of graphs with no two vertex-disjoint cycles, while Slilaty has given a structural characterisation of graphs with no two vertex-disjoint odd cycles; his result is in fact more general,…
A famous result by R\"odl, Ruci\'nski, and Szemer\'edi guarantees a (tight) Hamilton cycle in $k$-uniform hypergraphs $H$ on $n$ vertices with minimum $(k-1)$-degree $\delta_{k-1}(H)\geq (1/2+o(1))n$, thereby extending Dirac's result from…
The 3-Decomposition Conjecture states that every connected cubic graph can be decomposed into a spanning tree, a 2-regular subgraph and a matching. We show that this conjecture holds for the class of connected plane cubic graphs.
We study dynamics and bifurcations of two-dimensional reversible maps having non-transversal heteroclinic cycles containing symmetric saddle periodic points. We consider one-parameter families of reversible maps unfolding generally the…
We consider a family of toroidal graphs, denoted by $\mathcal{T}_{i, j}$, which contain neither $i$-cycles nor $j$-cycles. A graph $G$ is $(d, h)$-decomposable if it contains a subgraph $H$ with $\Delta(H) \leq h$ such that $G - E(H)$ is a…
Szepietowski [A. Szepietowski, Hamiltonian cycles in hypercubes with $2n-4$ faulty edges, Information Sciences, 215 (2012) 75--82] observed that the hypercube $Q_n$ is not Hamiltonian if it contains a trap disconnected halfway. A proper…
We study some properties of a serial (i.e. one-by-one) symmetric exchange of elements of two disjoint bases of a matroid. We show that any two elements of one base have a serial symmetric exchange with some two elements of the other base.…
We define an algebraic setup of homology for hypergraphs, which defaults to simplicial homology in the case of graphs, and study its basic properties. As part of our study we define algebraic spanning trees of hypergraphs, along with…
Bipartite graphs are a fundamental concept in graph theory with diverse applications. A graph is bipartite iff it contains no odd cycles, a characteristic that has many implications in diverse fields ranging from matching problems to the…
An odd (resp. even) subgraph in a multigraph is its subgraph in which every vertex has odd (resp. even) degree. We say that a multigraph can be decomposed into two odd subgraphs if its edge set can be partitioned into two sets so that both…
In this paper, we introduce two new forms of the dual Hartwig-Spindelb{\"o}ck decomposition and employ them to derive explicit representations for several classes of dual generalized inverses. Building on these representations, we further…
Las Vergnas and Hamidoune studied the number of circuits needed to determine an oriented matroid. In this paper we investigate this problem and some new variants, as well as their interpretation in particular classes of matroids. We present…
Graphings serve as limit objects for bounded-degree graphs. We define the ``cycle matroid'' of a graphing as a submodular setfunction, with values in [0,1], which generalizes (up to normalization) the cycle matroid of finite graphs. We…
We characterize which systems of sign vectors are the cocircuits of an oriented matroid in terms of the cocircuit graph.
We investigate the metric structure of the intersection lattice L(B(n,k)) of the discriminantal arrange ment using circuit supports. We show that the cover graph associated with L(B(n,k)) is isometrically embedded into a hypercube, making…