Mathematics - Combinatorics Publications (50)

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Mathematics - Combinatorics Publications

A graph is said to be well-dominated if all its minimal dominating sets are of the same size. The class of well-dominated graphs forms a subclass of the well studied class of well-covered graphs. We introduce the notion of an irreducible dominating set, a variant of dominating set generalizing both minimal dominating sets and minimal total dominating sets. Read More


The vertices of any graph with $m$ edges may be partitioned into two parts so that each part meets at least $\frac{2m}{3}$ edges. Bollob\'as and Thomason conjectured that the vertices of any $r$-uniform hypergraph with $m$ edges may likewise be partitioned into $r$ classes such that each part meets at least $\frac{r}{2r-1}m$ edges. In this paper we prove the weaker statement that, for each $r\ge 4$, a partition into $r$ classes may be found in which each class meets at least $\frac{r}{3r-4}m$ edges, a substantial improvement on previous bounds. Read More


We show that every $2$-connected cubic graph $G$ has a cycle double cover if $G$ has a spanning subgraph $F$ such that (i) every component of $F$ has an even number of vertices (ii) every component of $F$ is either an even cycle or a subdivision of a Kotzig graph and (iii) the components of $F$ are connected to each other in a certain general manner. Read More


It is shown that the family of generalized Petersen graphs can be recognized in linear time. Read More


We present an enumeration of orientably-regular maps with automorphism group isomorphic to the twisted linear fractional group $M(q^2)$ for any odd prime power $q$. Read More


Given $\eta \in [0, 1]$, a colouring $C$ of $V(G)$ is an $\eta$-majority colouring if at most $\eta d^+(v)$ out-neighbours of $v$ have colour $C(v)$, for any $v \in V(G)$. We show that every digraph $G$ equipped with an assignment of lists $L$, each of size at least $k$, has a $2/k$-majority $L$-colouring. For even $k$ this is best possible, while for odd $k$ the constant $2/k$ cannot be replaced by any number less than $2/(k+1)$. Read More


Strong external difference families (SEDFs) were introduced by Paterson and Stinson as a more restrictive version of external difference families. SEDFs can be used to produce optimal strong algebraic manipulation detection codes. We characterize the parameters $(v, m, k, \lambda)$ of a nontrivial SEDF that is near-complete (satisfying $v=km+1$). Read More


This paper presents a combinatorial construction of low-density parity-check (LDPC) codes from difference covering arrays. While the original construction by Gallagher was by randomly allocating bits in a sparse parity-check matrix, over the past 20 years researchers have used a variety of more structured approaches to construct these codes, with the more recent constructions of well-structured LDPC coming from balanced incomplete block designs (BIBDs) and from Latin squares over finite fields. However these constructions have suffered from the limited orders for which these designs exist. Read More


This paper studies graphs that have two tree decompositions with the property that every bag from the first decomposition has a bounded-size intersection with every bag from the second decomposition. We show that every graph in each of the following classes has a tree decomposition and a linear-sized path decomposition with bounded intersections: (1) every proper minor-closed class, (2) string graphs with a linear number of crossings in a fixed surface, (3) graphs with linear crossing number in a fixed surface. We then show that every $n$-vertex graph that has a tree decomposition and a linear-sized path decomposition with bounded intersections has $O(\sqrt{n})$ treewidth. Read More


For any positive integer $m$, the complete graph on $2^{2m}(2^m+2)$ vertices is decomposed into $2^m+1$ commuting strongly regular graphs, which give rise to a symmetric association scheme of class $2^{m+2}-2$. Furthermore, the eigenmatrices of the symmetric association schemes are determined explicitly. As an application, the eigenmatrix of the commutative strongly regular decomposition obtained from the strongly regular graphs is derived. Read More


We express the integral form Macdonald polynomials as a weighted sum of Shareshian and Wachs' chromatic quasisymmetric functions of certain graphs. Then we use known expansions of these chromatic quasisymmetric functions into Schur and power sum symmetric functions to provide Schur and power sum formulas for the integral form Macdonald polynomials. Since the (integral form) Jack polynomials are a specialization of integral form Macdonald polynomials, we obtain analogous formulas for Jack polynomials as corollaries. Read More


We investigate which graphs H have the property that in every graph with bounded clique number and sufficiently large chromatic number, some induced subgraph is isomorphic to a subdivision of H. In an earlier paper, one of us proved that every tree has this property; and in another earlier paper with M. Chudnovsky, we proved that every cycle has this property. Read More


Combinatorial discrepancy is a complexity measure of a collection of sets which quantifies how well the sets in the collection can be simultaneously balanced. More precisely, we are given an n-point set $P$, and a collection $\mathcal{F} = \{F_1, .. Read More


In this paper we investigate the number of integer points lying in dilations of lattice path matroid polytopes. We give a characterization of such points as polygonal paths in the diagram of the lattice path matroid. Furthermore, we prove that lattice path matroid polytopes are affinely equivalent to a family of distributive polytopes. Read More


We give two graph theoretical characterizations of tope graphs of (complexes of) oriented matroids. The first is in terms of excluded partial cube minors, the second is that all antipodal subgraphs are gated. A direct consequence is a third characterization in terms of zone graphs of tope graphs. Read More


Substantial efforts have been made to compute or estimate the minimum number $c(G)$ of cycles needed to partition the edges of an Eulerian graph. We give an equivalent characterization of Eulerian graphs of treewidth $2$ and with maximum degree $4$. This characterization enables us to present a linear time algorithm for the computation of $c(G)$ for all $G$ in this class. Read More


Laman graphs model planar frameworks that are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. Such realizations can be seen as solutions of systems of quadratic equations prescribing the distances between pairs of points. Read More


Motivated by applications in cancer genomics and following the work of Hajirasouliha and Raphael (WABI 2014), Hujdurovi\'{c} et al. (WABI 2015, full version to appear in IEEE TCBB) introduced the minimum conflict-free row split (MCRS) problem: split each row of a given binary matrix into a bitwise OR of a set of rows so that the resulting matrix corresponds to a perfect phylogeny and has the minimum number of rows among all matrices with this property. Hajirasouliha and Raphael also proposed the study of a similar problem, referred to as the minimum distinct conflict-free row split (MDCRS) problem, in which the task is to minimize the number of distinct rows of the resulting matrix. Read More


In this paper we study the art gallery problem, which is one of the fundamental problems in computational geometry. The objective is to place a minimum number of guards inside a simple polygon such that the guards together can see the whole polygon. We say that a guard at position $x$ sees a point $y$ if the line segment $xy$ is fully contained in the polygon. Read More


Bryant, Horsley, Maenhaut and Smith recently gave necessary and sufficient conditions for when the complete multigraph can be decomposed into cycles of specified lengths $m_1,m_2,\ldots,m_\tau$. In this paper we characterise exactly when there exists a packing of the complete multigraph with cycles of specified lengths $m_1,m_2,\ldots,m_\tau$. While cycle decompositions can give rise to packings by removing cycles from the decomposition, in general it is not known when there exists a packing of the complete multigraph with cycles of various specified lengths. Read More


A directed acyclic graph G = (V, E) is pseudo-transitive with respect to a given subset of edges E1, if for any edge ab in E1 and any edge bc in E, we have ac in E. We give algorithms for computing longest chains and demonstrate geometric applications that unify and improves some important past results. (For specific applications see the introduction. Read More


The generators of the classical Specht module satisfy intricate relations. We introduce the Specht matroid, which keeps track of these relations, and the Specht polytope, which also keeps track of convexity relations. We establish basic facts about the Specht polytope, for example, that the symmetric group acts transitively on its vertices and irreducibly on its ambient real vector space. Read More


We determine the maximum distance between any two of the center, centroid, and subtree core among trees with a given order. Corresponding results are obtained for trees with given maximum degree and also for trees with given diameter. The problem of the maximum distance between the centroid and the subtree core among trees with given order and diameter becomes difficult. Read More


In 2007, Dmytrenko, Lazebnik and Williford posed two related conjectures about polynomials over finite fields. Conjecture~1 is a claim about the uniqueness of certain monomial graphs. Conjecture~2, which implies Conjecture~1, deals with certain permutation polynomials of finite fields. Read More


We present a new family of one-coincidence sequence sets suitable for frequency hopping code division multiple access (FH-CDMA) systems with dispersed (low density) sequence elements. These sets are derived from one-coincidence prime sequence sets, such that for each one-coincidence prime sequence set there is a new one-coincidence set comprised of sequences with dispersed sequence elements, required in some circumstances, for FH-CDMA systems. Getting rid of crowdedness of sequence elements is achieved by doubling the size of the sequence element alphabet. Read More


Let $Q$ be a free Boltzmann quadrangulation with simple boundary decorated by a critical ($p=3/4$) face percolation configuration. We prove that the chordal percolation exploration path on $Q$ between two marked boundary edges converges in the scaling limit to chordal SLE$_6$ on an independent $\sqrt{8/3}$-Liouville quantum gravity disk (equivalently, a Brownian disk). The topology of convergence is the Gromov-Hausdorff-Prokhorov-uniform topology, the natural analog of the Gromov-Hausdorff topology for curve-decorated metric measure spaces. Read More


We prove that the free Boltzmann quadrangulation with simple boundary and fixed perimeter, equipped with its graph metric, natural area measure, and the path which traces its boundary converges in the scaling limit to the free Boltzmann Brownian disk. The topology of convergence is the so-called Gromov-Hausdorff-Prokhorov-uniform (GHPU) topology, the natural analog of the Gromov-Hausdorff topology for curve-decorated metric measure spaces. From this we deduce that a random quadrangulation of the sphere decorated by a $2l$-step self-avoiding loop converges in law in the GHPU topology to the random curve-decorated metric measure space obtained by gluing together two Brownian disks along their boundaries. Read More


A new tropical plactic algebra is introduced in which the Knuth relations are inferred from the underlying semiring arithmetics, encapsulating the ubiquitous plactic monoid $\mathcal{P}_n$. This algebra manifests a natural framework for accommodating representations of $\mathcal{P}_n$, or equivalently of Young tableaux, and its moderate coarsening -- the cloaktic monoid $\mathcal{K}_n$ and the co-cloaktic $ ^{\operatorname{co}}\mathcal{K}_n$. The faithful linear representations of $\mathcal{K}_n$ and $\, ^{\operatorname{co}} \mathcal{K}_n$ by tropical matrices, which constitute a tropical plactic algebra, are shown to provide linear representations of the plactic monoid. Read More


It is well-known that the Pachner graph of triangulated n-vertex 2-spheres is connected, i.e., each pair of n-vertex 2-spheres can be turned into each other by a sequence of edge flips. Read More


A subfamily $\{F_1,F_2,\dots,F_{|P|}\}\subseteq \mathcal{F}$ of sets is a copy of a poset $P$ in $\mathcal{F}$ if there exists a bijection $\phi:P\rightarrow \{F_1,F_2,\dots,F_{|P|}\}$ such that whenever $x \le_P x'$ holds, then so does $\phi(x)\subseteq \phi(x')$. For a family $\mathcal{F}$ of sets, let $c(P,\mathcal{F})$ denote the number of copies of $P$ in $\mathcal{F}$, and we say that $\mathcal{F}$ is $P$-free if $c(P,\mathcal{F})=0$ holds. For any two posets $P,Q$ let us denote by $La(n,P,Q)$ the maximum number of copies of $Q$ over all $P$-free families $\mathcal{F} \subseteq 2^{[n]}$, i. Read More


We provide simple necessary and sufficient conditions for the existence of a standard Young tableau of a given shape and major index $r$ mod $n$, for all $r$. Our result generalizes the $r=1$ case due essentially to (1974) and proves a recent conjecture due to Sundaram (2016) for the $r=0$ case. A byproduct of the proof is an asymptotic equidistribution result for "almost all" shapes. Read More


This paper explores the orbit structure and homomesy (constant averages over orbits) properties of certain actions of toggle groups on the collection of independent sets of a path graph. In particular we prove a generalization of a homomesy conjecture of Propp that for the action of a "Coxeter element" of vertex toggles, the difference of indicator functions of symmetrically-located vertices is 0-mesic. Then we use our analysis to show facts about orbit sizes that are easy to conjecture but nontrivial to prove. Read More


We treat problems of fair division, their various interconnections, and their relations to Sperner's lemma and the KKM theorem as well as their variants. We prove extensions of Alon's necklace splitting result in certain regimes and relate it to hyperplane mass partitions. We show the existence of fair cake division and rental harmony in the sense of Su even in the absence of full information. Read More


The Erd\H{o}s-Ginzburg-Ziv constant of an abelian group $G$, denoted $\mathfrak{s}(G)$, is the smallest $k\in\mathbb{N}$ such that any sequence of length $\mathfrak{s}(G)$ contains a zero-sum subsequence of length $\exp(G)$. In this paper, we use the multi-slice rank method, which is a variant of Tao's slice rank method, to prove that \[ \mathfrak{s}\left(\mathbb{F}_{p}^{n}\right)\leq3(p-1)p!\left(J(p)p\right)^{n} \] where $0.8414Read More


`Double edge swaps' transform one graph into another while preserving the graph's degree sequence, and have thus been used in a number of popular Markov chain Monte Carlo (MCMC) sampling techniques. However, while double edge-swap MCMC sampling can, for any fixed degree sequence, sample simple graphs, multigraphs, and pseudographs uniformly, this is not true for graphs which allow self-loops but not multiedges (loopy graphs). Indeed, we exactly characterize the degree sequences where double edge swaps cannot reach every valid loopy graph and develop an efficient algorithm to determine such degree sequences. Read More


We consider a problem introduced by Mossel and Ross [Shotgun assembly of labeled graphs, arXiv:1504.07682]. Suppose a random $n\times n$ jigsaw puzzle is constructed by independently and uniformly choosing the shape of each "jig" from $q$ possibilities. Read More


In 2010, Joyce et. al defined the leverage centrality of vertices in a graph as a means to analyze functional connections within the human brain. In this metric a degree of a vertex is compared to the degrees of all it neighbors. Read More


We introduce new natural generalizations of the classical descent and inversion statistics for permutations, called width-$k$ descents and width-$k$ inversions. These variations induce generalizations of the excedance and major statistics, providing a framework in which the most well-known equidistributivity results for classical statistics are paralleled. We explore additional relationships among the statistics providing specific formulas in certain special cases. Read More


Many important problems in combinatorics and other related areas can be phrased in the language of independent sets in hypergraphs. Recently Balogh, Morris and Samotij, and independently Saxton and Thomason developed very general container theorems for independent sets in hypergraphs; both of which have seen numerous applications to a wide range of problems. In this paper we use the container method to prove results that correspond to problems concerning tuples of disjoint independent sets in hypergraphs. Read More


We use the rationality of the generalized $h^{th}$ convergent functions, $Conv_h(\alpha, R; z)$, to the infinite J-fraction expansions enumerating the generalized factorial product sequences, $p_n(\alpha, R) = R(R+\alpha)\cdots(R+(n-1)\alpha)$, defined in the references to construct new congruences and $h$-order finite difference equations for generalized factorial functions modulo $h \alpha^t$ for any primes or odd integers $h \geq 2$ and integers $0 \leq t \leq h$. Special cases of the results we consider within the article include applications to new congruences and exact formulas for the $\alpha$-factorial functions, $n!_{(\alpha)}$. Applications of the new results we consider within the article include new finite sums for the $\alpha$-factorial functions, restatements of classical necessary and sufficient conditions of the primality of special integer subsequences and tuples, and new finite sums for the single and double factorial functions modulo integers $h \geq 2$. Read More


With any (not necessarily proper) edge $k$-colouring $\gamma:E(G)\longrightarrow\{1,\dots,k\}$ of a graph $G$,one can associate a vertex colouring $\sigma\_{\gamma}$ given by $\sigma\_{\gamma}(v)=\sum\_{e\ni v}\gamma(e)$.A neighbour-sum-distinguishing edge $k$-colouring is an edge colouring whose associated vertex colouring is proper.The neighbour-sum-distinguishing index of a graph $G$ is then the smallest $k$ for which $G$ admitsa neighbour-sum-distinguishing edge $k$-colouring. Read More


In 1972, Erd\"{o}s - Faber - Lov\'{a}sz (EFL) conjectured that, if $\textbf{H}$ is a linear hypergraph consisting of $n$ edges of cardinality $n$, then it is possible to color the vertices with $n$ colors so that no two vertices with the same color are in the same edge. In 1978, Deza, Erd\"{o}s and Frankl had given an equivalent version of the same for graphs: Let $G= \bigcup_{i=1}^{n} A_i$ denote a graph with $n$ complete graphs $A_1, A_2,$ $ \dots , A_n$, each having exactly $n$ vertices and have the property that every pair of complete graphs has at most one common vertex, then the chromatic number of $G$ is $n$. The clique degree $d^K(v)$ of a vertex $v$ in $G$ is given by $d^K(v) = |\{A_i: v \in V(A_i), 1 \leq i \leq n\}|$. Read More


In the present paper we consider point distributions in compact connected two-point homogeneous spaces (Riemannian symmetric spaces of rank one). All such spaces are known, they are spheres in the Euclidean spaces, the real, complex and quaternionic projective spaces and the octonionic projective plane. Our concern is with discrepancies of distributions in metric balls and sums of pairwise distances between points of distributions for all such spaces. Read More


In this paper, we investigate the problem of separating a set $X$ of points in $\mathbb{R}^{2}$ with an arrangement of $K$ lines such that each cell contains an asymptotically equal number of points (up to a constant ratio). We consider a property of curves called the stabbing number, defined to be the maximum countable number of intersections possible between the curve and a line in the plane. We show that large subsets of $X$ lying on Jordan curves of low stabbing number are an obstacle to equal separation. Read More


Consider a surface $S$ and let $M\subset S$. If $S\setminus M$ is not connected, then we say $M$ \emph{separates} $S$, and we refer to $M$ as a \emph{separating set} of $S$. If $M$ separates $S$, and no proper subset of $M$ separates $S$, then we say $M$ is a \emph{minimal separating set} of $S$. Read More


It is well known that each nonnegative integral flow on a graph can be decomposed into a sum of nonnegative graphic circuit flows, which cannot be further decomposed into nonnegative integral sub-flows. This is equivalent to saying that the indecomposable flows on graphs are those graphic circuit flows. Turning from graphs to signed graphs, the indecomposable flows are much richer than those of unsigned graphs. Read More


Weingarten calculus is a completely general and explicit method to compute the moments of the Haar measure on compact subgroups of matrix algebras. Particular cases of this calculus were initiated by theoretical physicists -- including Weingarten, after whom this calculus was coined by the first author, after investigating it systematically. Substantial progress was achieved subsequently by the second author and coworkers, based on representation theoretic and combinatorial techniques. Read More


Let $\mathcal{S}\subset\mathbb{F}_{q}^{n}$ be the subset of self-orthogonal vectors in $\mathbb{F}_q^n$, which has size $|\mathcal{S}|=q^{n-1}+O\left(\sqrt{q^{n}}\right)$. In this paper we use the recently developed slice rank method to show that any set $E\subset\mathcal{S}$ of size \[ |E|>(n+1)^{(q-1)(k-1)} \] contains a $k$-tuple of distinct mutually orthogonal vectors, where $n\geq\binom{k}{2}$ and $q=p^{r}$ with $p\geq k$. The key innovation is a more general version of the slice rank of a tensor, which we call the multi-slice rank. Read More


We formulate three current models of discrete-time quantum walks in a combinatorial way. These walks are shown to be closely related to rotation systems and 1-factorizations of graphs. For two of the models, we compute the traces and total entropies of the average mixing matrices for some cubic graphs. Read More


We introduce a class of fixed points of primitive morphisms among aperiodic binary generalized pseudostandard words. We conjecture that this class contains all fixed points of primitive morphisms among aperiodic binary generalized pseudostandard words that are not standard Sturmian words. Read More