Combinatorics (math.CO)

Abstract: Villarceau torus is a discrete graph theory model of spiral torus which is called Helical Toroidal Electron Model in Physics. It also represents the double stranded helix model of DNA. The spiral torus or toroidal helix in Physics and Molecular Biology is continuous whereas the Villarceau torus is discrete. The main contribution of the paper is the identification of cycles which are equivalent to toroidal helix in spiral torus. In addition, the poloidal revolution and the toroidal revolutions of Villarceau torus are computed. The paper identifies some striking differences between ring torus and Villarceau torus. It is proved that Villarceau torus does not admit any convex edgecuts and convex cycles other than 4-cycles. The convex cut method is extended to graphs which do not admit convex edgecuts. Using this technique, the network distance (Wiener Index) is computed, Also an optimal congestion-balanced routing for Villarceau torus is designed.

Abstract: In this note we prove: {\it Let $D$ be a 2-strong digraph of order $n$ such that its $n-1$ vertices have degrees at least $n+k$ and the remaining vertex $z$ has degree at least $n-k-4$, where $k$ is a positive integer. If $D$ contains a cycle of length at least $n-k-2$ passing through $z$, then $D$ is Hamiltonian}.

Abstract: This article introduces the package MatchingFields for Macaulay2 and highlights some open problems. A matching field is a combinatorial object whose data encodes a candidate toric degeneration of a Grassmannian or partial flag variety of type A. Each coherent matching field is associated to a certain maximal cone of the respective tropical variety. The MatchingFields package provides methods to construct matching fields along with their rings, ideals, polyhedra and matroids. The package also supplies methods to test whether a matching field is coherent, linkage and gives rise to a toric degeneration.

Abstract: Given $n\in k\mathbb{N}$ elements set $V$ and $k$-uniform hypergraphs $\mathcal{H}_1,\ldots,\mathcal{H}_{n/k}$ on $V$. A rainbow perfect matching is a collection of pairwise disjoint edges $E_1\in \mathcal{H}_1,\ldots,E_{n/k}\in \mathcal{H}_{n/k}$ such that $E_1\cup\cdots\cup E_{n/k}=V$. In this paper, we determine the minimum $\ell$-degree condition that guarantees the existence of a rainbow perfect matching for sufficiently large $n$ and $\ell\geq k/2$.

Abstract: A conjecture of Kalai asserts that for $d\geq 4$, the affine type of a prime simplicial $d$-polytope $P$ can be reconstructed from the space of affine $2$-stresses of $P$. We prove this conjecture for all $d\geq 5$. We also prove the following generalization: for all pairs $(i,d)$ with $2\leq i\leq \lceil \frac d 2\rceil-1$, the affine type of a simplicial $d$-polytope $P$ that has no missing faces of dimension $\geq d-i+1$ can be reconstructed from the space of affine $i$-stresses of $P$. A consequence of our proofs is a strengthening of the Generalized Lower Bound Theorem: for any simplicial $(d-1)$-sphere $\Delta$ and $1\leq k\leq \lceil\frac{d}{2}\rceil-1$, $g_k(\Delta)$ is at least as large as the number of missing $(d-k)$-faces of $\Delta$; furthermore, for $1\leq k\leq \lfloor\frac{d}{2}\rfloor-1$, equality holds if and only if $\Delta$ is $k$-stacked. Finally, we show that for $d\geq 4$, any simplicial $d$-polytope $P$ that has no missing faces of dimension $\geq d-1$ is redundantly rigid, that is, for each edge $e$ of $P$, there exists an affine $2$-stress on $P$ with a non-zero value on $e$.

Abstract: Esperet and Joret proved that planar graphs with bounded maximum degree are 3-colorable with bounded clustering. Liu and Wood asked whether the conclusion holds with the assumption of the bounded maximum degree replaced by assuming that no two vertices have many common neighbors. We answer this question in positive, in the following stronger form: Let P''_t be the complete join of two isolated vertices with a path on t vertices. For any surface Sigma, a subgraph-closed class of graphs drawn on Sigma is 3-choosable with bounded clustering if and only if there exists t such that P''_t does not belong to the class.

Abstract: Let $S$ be a subset of a group $G$ (not necessarily abelian) such that $S\,\cap -S$ is empty or contains only elements of order $2$, and let $\mathbf{h}=(h_1,\ldots, h_m)\in \mathbb{N}^m$ and $\mathbf{k}=(k_1, \ldots, k_n)\in \mathbb{N}^n$. A generalized Heffter array GHA$^{\lambda}_S(m, n; \mathbf{h}, \mathbf{k})$ over $G$ is an $m\times n$ matrix $A=(a_{ij})$ such that: the $i$-th row (resp. $j$-th column) of $A$ contains exactly $h_i$ (resp. $k_j$) nonzero elements, and the list $\{a_{ij}, -a_{ij}\mid a_{ij}\neq 0\}$ equals $\lambda$ times the set $S\,\cup\, -S$. We speak of a zero sum (resp. nonzero sum) GHA if each row and each column of $A$ sums to zero (resp. a nonzero element), with respect to some ordering. In this paper, we use near alternating sign matrices to build both zero and nonzero sum GHAs, over cyclic groups, having the further strong property of being simple. In particular, we construct zero sum and simple GHAs whose row and column weights are congruent to $0$ modulo $4$. Furthermore, we build nonzero sum GHA$^{\lambda}_S(m, n; \mathbf{h}, \mathbf{k})$ over an arbitrary group $G$ whenever $S$ contains enough noninvolutions, thus extending previous nonconstructive results where $\pm S = G\setminus H$ for some subgroup $H$ of $G$. Finally, we describe how GHAs can be used to build orthogonal decompositions and biembeddings of Cayley graphs (over groups not necessarily abelian) onto orientable surfaces.

Abstract: A subset $S$ of vertices of a graph $G$ is a \emph{general position set} if no shortest path in $G$ contains three or more vertices of $S$. In this paper, we generalise a problem of M. Gardner to graph theory by introducing the \emph{lower general position number} $\gp ^-(G)$ of $G$, which is the number of vertices in a smallest maximal general position set of $G$. We show that ${\rm gp}^-(G) = 2$ if and only if $G$ contains a universal line and determine this number for several classes of graphs, including Kneser graphs $K(n,2)$, line graphs of complete graphs, and Cartesian and direct products of two complete graphs. We also prove several realisation results involving the lower general position number, the general position number and the geodetic number, and compare it with the lower version of the monophonic position number. We provide a sharp upper bound on the size of graphs with given lower general position number. Finally we demonstrate that the decision version of the lower general position problem is NP-complete.

Abstract: Necessary and sufficient conditions on $A,B\in \mathbb{F}_{q^3}^*$ for $f(X)=X^{q^2-q+1}+AX^{q^2}+BX$ being a permutation polynomial of $\mathbb{F}_{q^3}$ are investigated via a connection with algebraic varieties over finite fields.