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Tilings in randomly perturbed graphs: bridging the gap between Hajnal-Szemerédi and Johansson-Kahn-VuApr 22 2019A perfect $K_r$-tiling in a graph $G$ is a collection of vertex-disjoint copies of $K_r$ that together cover all the vertices in $G$. In this paper we consider perfect $K_r$-tilings in the setting of randomly perturbed graphs; a model introduced by Bohman, ... More
The weak separation in higher dimensionsApr 22 2019For an odd integer $r>0$ and an integer $n>r$, we introduce a notion of weakly $r$-separated subsets of $[n]=\{1,2,\ldots,n\}$. When $r=1$, this corresponds to the concept of weak separation introduced and studied by Leclerc and Zelevinsky. In this paper, ... More
Transpositional sequences and multigraphsApr 22 2019When ${\bf t} := \langle t_1,t_2,\ldots,t_k\rangle$ is a sequence of transpositions on the finite set $n:=\{0,1,\ldots,n-1\}$, then $\bigcirc{\bf t}:= t_1\circ t_2\circ\cdots\circ t_k$ denotes the compositional product of the sequence. Our paper treats ... More
K-theoretic crystals for set-valued tableaux of rectangular shapesApr 21 2019In earlier work with C. Monical (2018), we introduced the notion of a K-crystal, with applications to K-theoretic Schubert calculus and the study of Lascoux polynomials. We conjectured that such a K-crystal structure existed on the set of semistandard ... More
Linear codes over signed graphsApr 20 2019We give formulas, in terms of graph theoretical invariants, for the minimum distance and the generalized Hamming weights of the linear code generated by the rows of the incidence matrix of a signed graph over a finite field, and for those of its dual ... More
Chromatic symmetric functions in noncommuting variables revisitedApr 19 2019In 1995 Stanley introduced a generalization of the chromatic polynomial of a graph $G$, called the chromatic symmetric function, $X_G$, which was generalized to noncommuting variables, $Y_G$, by Gebhard-Sagan in 2001. Recently there has been a renaissance ... More
A survey on maximal green sequencesApr 19 2019Maximal green sequences appear in the study of Fomin-Zelevinsky's cluster algebras. They are useful for computing refined Donaldson-Thomas invariants, constructing twist automorphisms and proving the existence of theta bases and generic bases. We survey ... More
New results on pseudosquare avoidanceApr 19 2019We start by considering binary words containing the minimum possible numbers of squares and antisquares (where an antisquare is a word of the form $x \overline{x}$), and we completely classify which possibilities can occur. We consider avoiding $x p(x)$, ... More
The Douglas-Rachford Algorithm for Convex and Nonconvex Feasibility ProblemsApr 19 2019The Douglas-Rachford method, a projection algorithm designed to solve continuous optimization problems, forms the basis of a useful heuristic for solving combinatorial optimization problems. In order to successfully use the method, it is necessary to ... More
k-Spectra of c-Balanced WordsApr 19 2019A word $u$ is a scattered factor of $w$ if $u$ can be obtained from $w$ by deleting some of its letters. That is, there exist the (potentially empty) words $u_1,u_2,..., u_n$, and $v_0,v_1,..,v_n$ such that $u = u_1u_2...u_n$ and $w = v_0u_1v_1u_2v_2...u_nv_n$. ... More
A corollary of Stanley's Hook Content FormulaApr 18 2019We use the hook lengths of a partition to define two rectangular tableaux. We prove these tableaux have equal multisets of entries, first by elementary combinatorial arguments, and then using Stanley's Hook Content Formula and symmetric polynomials.
Generalized threshold arrangementsApr 18 2019An arrangement of hyperplanes is a finite collection of hyperplanes in a real Euclidean space. To such a collection one associates the characteristic polynomial that encodes the combinatorics of intersections of the hyperplanes. Finding the characteristic ... More
A $q$-analogue of Wilson's congruenceApr 18 2019Let ${\mathcal C}_n$ be the set of all permutation cycles of length $n$ over $\{1,2,\ldots,n\}$. Let $${\mathfrak f}_n(q):=\sum_{\sigma\in{\mathcal C}_{n+1}}q^{{\mathrm maj}\,\sigma} $$ be a $q$-analogue of the factorial $n!$, where ${\mathrm maj}$ denotes ... More
Planar Point Sets Determine Many Pairwise Crossing SegmentsApr 18 2019We show that any set of $n$ points in general position in the plane determines $n^{1-o(1)}$ pairwise crossing segments. The best previously known lower bound, $\Omega\left(\sqrt n\right)$, was proved more than 25 years ago by Aronov, Erd\H os, Goddard, ... More
Perfect State Transfer on Weighted Graphs of the Johnson SchemeApr 18 2019We characterize perfect state transfer on real-weighted graphs of the Johnson scheme $\mathcal{J}(n,k)$. Given $\mathcal{J}(n,k)=\{A_1, A_2, \cdots, A_k\}$ and $A(X) = w_0A_0 + \cdots + w_m A_m$, we show, using classical number theory results, that $X$ ... More
Partial associativity and rough approximate groupsApr 18 2019Suppose that a binary operation $\circ$ on a finite set $X$ is injective in each variable separately and also associative. It is easy to prove that $(X,\circ)$ must be a group. In this paper we examine what happens if one knows only that a positive proportion ... More
Upper bounds for bar visibility of subgraphs and n-vertex graphsApr 18 2019A $t$-bar visibility representation of a graph assigns each vertex up to $t$ horizontal bars in the plane so that two vertices are adjacent if and only if some bar for one vertex can see some bar for the other via an unobstructed vertical channel of positive ... More
An extremal problem for integer sparse recoveryApr 18 2019Motivated by the problem of integer sparse recovery we study the following question. Let $A$ be an $m \times d$ integer matrix whose entries are in absolute value at most $k$. How large can be $d=d(m,k)$ if all $m \times m$ submatrices of $A$ are non-degenerate? ... More
Maximal co-cliques in the Kneser graph on plane-solid flags in $PG(6,q)$Apr 18 2019For $q>27$ we determine the independence number $\alpha(\Gamma)$ of the Kneser graph $\Gamma$ on plane-solid flags in $PG(6,q)$. More precisely we describe all maximal independent sets of size at least $q^{11}$ and show that every other maximal example ... More
The hafnian of Toeplitz matrices of a special type, perfect matchings and Bessel polynomialsApr 18 2019We present a simple and convenient analytical formula for efficient exact computation of the hafnian of Toeplitz matrices of a special type. An interpretation of the obtained results is given in the language of perfect matchings and Bessel polynomials. ... More
Approaching Cusick's conjecture on the sum-of-digits functionApr 18 2019Cusick's conjecture on the binary sum of digits $s(n)$ of a nonnegative integer $n$ states the following: for all nonnegative integers $t$ we have \[ c_t=\lim_{N\rightarrow\infty}\frac 1N\left\lvert\{n<N:s(n+t)\geq s(n)\}\right\rvert>1/2. \] We prove ... More
Monochromatic disconnection: Erdős-Gallai-type problems and product graphsApr 18 2019For an edge-colored graph $G$, we call an edge-cut $M$ of $G$ monochromatic if the edges of $M$ are colored with a same color. The graph $G$ is called monochromatically disconnected if any two distinct vertices of $G$ are separated by a monochromatic ... More
The Hopf Monoid of Orbit PolytopesApr 17 2019Many families of combinatorial objects have a Hopf monoid structure. Aguiar and Ardila introduced the Hopf monoid of generalized permutahedra and showed that it contains various other notable combinatorial families as Hopf submonoids, including graphs, ... More
Re-pairing bracketsApr 17 2019Consider the following one-player game. Take a well-formed sequence of opening and closing brackets. As a move, the player can pair any opening bracket with any closing bracket to its right, erasing them. The goal is to re-pair (erase) the entire sequence, ... More
Hook-length formula and applications to alternating permutationsApr 17 2019In this paper, we take interest in finding applications for a hook-length formula recently proved in (Morales Pak Panova 2016). This formula can be applied to give a non trivial relation between alternating permutations and weighted Dyck paths. First, ... More
Type $\tilde{C}$ Temperley-Lieb algebra quotients and Catalan combinatoricsApr 17 2019We study some algebraic and combinatorial features of two algebras that arise as quotients of Temperley-Lieb algebras of type $\tilde{C}$, namely, the two-boundary Temperley-Lieb algebra and the symplectic blob algebra. We provide a monomial basis for ... More
A modification of Wythoff's NimApr 17 2019We modify Wythoff's game by allowing an additional move. The $P$-positions in our game can be derived from the table of letter positions in the Tribonacci word. This is related to the recent solution of the Greedy Queens in a spiral problem. Our analysis ... More
Higher dimensional connectivity and minimal degree of random graphs with an eye towards minimal free resolutionsApr 17 2019In this note we define and study graph invariants generalizing to higher dimension the maximum degree of a vertex and the vertex-connectivity (our $0$-dimensional cases). These are known to coincide almost surely in any regime for Erdoes-Renyi random ... More
On the neighborhood complex of $\vec{s}$-stable Kneser graphsApr 17 2019In 2002, A. Bj\"orner and M. de Longueville showed the neighborhood complex of the $2$-stable Kneser graph ${KG(n, k)}_{2-\textit{stab}}$ has the same homotopy type as the $(n-2k)$-sphere. A short time ago, an analogous result about the homotopy type ... More
Upper tails via high moments and entropic stabilityApr 17 2019Suppose that $X$ is a bounded-degree polynomial with nonnegative coefficients on the $p$-biased discrete hypercube. Our main result gives sharp estimates on the logarithmic upper tail probability of $X$ whenever an associated extremal problem satisfies ... More
Circularly squarefree words and unbordered conjugates: a new approachApr 17 2019Using a new approach based on automatic sequences, logic, and a decision procedure, we reprove some old theorems about circularly squarefree words and unbordered conjugates in a new and simpler way. Furthermore, we prove three new results about unbordered ... More
The intersection of two vertex coloring problemsApr 17 2019A hole is an induced cycle with at least four vertices. A hole is even if its number of vertices is even. Given a set L of graphs, a graph G is L-free if G does not contain any graph in L as an induced subgraph. Currently, the following two problems are ... More
A unified proof of conjectures on cycle lengths in graphsApr 17 2019In this paper, we prove a tight minimum degree condition in general graphs for the existence of paths between two given endpoints, whose lengths form a long arithmetic progression with common difference one or two. This allows us to obtain a number of ... More
A Lower Bound for Relaxed Locally Decodable CodesApr 17 2019A locally decodable code (LDC) C:{0,1}^k -> {0,1}^n is an error correcting code wherein individual bits of the message can be recovered by only querying a few bits of a noisy codeword. LDCs found a myriad of applications both in theory and in practice, ... More
The LexCycle on $\overline{P_{2}\cup P_{3}}$-free Cocomparability GraphsApr 17 2019A graph $G$ is a cocomparability graph if there exists an acyclic transitive orientation of the edges of its complement graph $\overline{G}$. Starting at some ordering $\sigma_{0}$ of $G$, let $\{\sigma_{i}\}_{i\geq 1}$ be the sequence of orderings such ... More
Determining Finite Connected Graphs Along the Quadratic Embedding Constants of PathsApr 17 2019The QE constant of a finite connected graph $G$, denoted by $\mathrm{QEC}(G)$, is by definition the maximum of the quadratic function associated to the distance matrix on a certain sphere of codimension two. We prove that the QE constants of paths $P_n$ ... More
Outliers in spectrum of sparse Wigner matricesApr 16 2019In this paper, we study the effect of sparsity on the appearance of outliers in the semi-circular law. Let $(W_n)_{n=1}^\infty$ be a sequence of random symmetric matrices such that each $W_n$ is $n\times n$ with i.i.d entries above and on the main diagonal ... More
Graded Quivers, Generalized Dimer Models and Toric GeometryApr 16 2019The open string sector of the topological B-model model on CY $(m+2)$-folds is described by $m$-graded quivers with superpotentials. This correspondence extends to general $m$ the well known connection between CY $(m+2)$-folds and gauge theories on the ... More
Extractors for small zero-fixing sourcesApr 16 2019A random variable $X$ is an $(n,k)$-zero-fixing source if for some subset $V\subseteq[n]$, $X$ is the uniform distribution on the strings $\{0,1\}^n$ that are zero on every coordinate outside of $V$. An $\epsilon$-extractor for $(n,k)$-zero-fixing sources ... More
On the cop number of toroidal graphsApr 16 2019We show that the cop number of toroidal graphs is at most 3. This resolves a conjecture by Schroeder from 2001 which is implicit in a question by Andreae from 1986.
$e$-Positivity Results and ConjecturesApr 16 2019In a 2016 ArXiv posting F. Bergeron listed a variety of symmetric functions $G[X;q]$ with the property that $G[X;1+q]$ is $e$-positive. A large subvariety of his examples could be explained by the conjecture that the Dyck path LLT polynomials exhibit ... More
Distribution of determinant of sum of matricesApr 16 2019Let $\mathbb{F}_q$ be an arbitrary finite field of order $q$. In this article, we study $\det S$ for certain types of subsets $S$ in the ring $M_2(\mathbb F_q)$ of $2\times 2$ matrices with entries in $\mathbb F_q$. For $i\in \mathbb{F}_q$, let $D_i$ ... More
Toric degenerations of flag varieties from matching field tableauxApr 16 2019We present families of tableaux which interpolate between the classical semi-standard Young tableaux and matching field tableaux. Algebraically, this corresponds to SAGBI bases of Pl\"ucker algebras. We show that each such family of tableaux leads to ... More
On the size of $(K_t,\mathcal{T}_k)$-co-critical graphsApr 16 2019Given an integer $r\ge1$ and graphs $G, H_1, \ldots, H_r$, we write $G \rightarrow ({H}_1, \ldots, {H}_r)$ if every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H_i$ in color $i$ for some $i\in\{1, \ldots, r\}$. A non-complete graph ... More
Spanning trees in complete bipartite graphs and resistance distance in nearly complete bipartite graphsApr 16 2019Using the theory of electrical network, we first obtain a simple formula for the number of spanning trees of a complete bipartite graph containing a certain matching or a certain tree. Then we apply the effective resistance (i.e., resistance distance ... More
p-Adic scaled space filling curve indices for high dimensional dataApr 16 2019Space filling curves are widely used in Computer Science. In particular Hilbert curves and their generalisations to higher dimension are used as an indexing method because of their nice locality properties. This article generalises this concept to the ... More
On the Chromatic Polynomial and Counting DP-ColoringsApr 16 2019The chromatic polynomial of a graph $G$, denoted $P(G,m)$, is equal to the number of proper $m$-colorings of $G$. The list color function of graph $G$, denoted $P_{\ell}(G,m)$, is a list analogue of the chromatic polynomial that has been studied since ... More
Semi-equivelar maps of Euler characteristics -2 with few verticesApr 16 2019We enumerate and classify all the semi equivelar maps on the surface of $ \chi=-2 $ with up to 12 vertices. We also determine which of these are vertex-transitive and which are not.
On the inducibility problem for random Cayley graphs of abelian groups with a few deleted verticesApr 16 2019Given a $k$-vertex graph $H$ and an integer $n$, what are the $n$-vertex graphs with the maximum number of induced copies of $H$? This question is closely related to the inducibility problem introduced by Pippenger and Golumbic in 1975, which asks for ... More
Sequences in Dihedral Groups with Distinct Partial ProductsApr 16 2019Given a subset $S$ of the non-identity elements of the dihedral group of order $2m$, is it possible to order the elements of $S$ so that the partial products are distinct? This is equivalent to the sequenceability of the group when $|S| = 2m-1$ and so ... More
Higher weight spectra of Veronese codesApr 16 2019We study q-ary linear codes C obtained from Veronese surfaces over finite fields. We show how one can find the higher weight spectra of these codes, or equivalently, the weight distribution of all extension codes of C over all field extensions of the ... More
Bootstrapping partition regularity of linear systemsApr 16 2019Apr 17 2019Suppose that $A$ is a $k \times d$ matrix of integers and write $\mathfrak{R}_A:\mathbb{N} \rightarrow \mathbb{N}\cup \{ \infty\}$ for the function taking $r$ to the largest $N$ such that there is an $r$-colouring $\mathcal{C}$ of $[N]$ with $\bigcup_{C ... More
Bootstrapping partition regularity of linear systemsApr 16 2019Suppose that $A$ is a $k \times d$ matrix of integers and write $\mathfrak{R}_A:\mathbb{N} \rightarrow \mathbb{N}\cup \{ \infty\}$ for the function taking $r$ to the largest $N$ such that there is an $r$-colouring $\mathcal{C}$ of $[N]$ with $\bigcup_{C ... More
On the Ramsey number of the Brauer configurationApr 16 2019We obtain a double exponential bound in Brauer's generalisation of van der Waerden's theorem, which concerns progressions monochromatic with their common difference. Such a result has been obtained independently and in much greater generality by Sanders. ... More
One-adhesive polymatroidsApr 16 2019Adhesive polymatroids were defined by F. Mat\'u\v{s} motivated by entropy functions. Two polymatroids are adhesive if they can be glued together along their joint part in a modular way; and are one-adhesive, if one of them has a single point outside their ... More
Quaternary Hermitian linear complementary dual codesApr 16 2019The largest minimum weights among quaternary Hermitian linear complementary dual codes are known for dimension $2$. In this paper, we give some conditions for the nonexistence of quaternary Hermitian linear complementary dual codes with large minimum ... More
A Turán-type theorem for large-distance graphs in Euclidean spaces, and related isodiametric problemsApr 16 2019Given a measurable set $A\subset \mathbb R^d$ we consider the "large-distance graph" $\mathcal{G}_A$, on the ground set $A$, in which each pair of points from $A$ whose distance is bigger than 2 forms an edge. We consider the problems of maximizing the ... More
The phylogeny number in the aspect of triangles and diamonds of a graphApr 16 2019Given an acyclic digraph $D$, the competition graph of $D$, denoted by $C(D)$, is the simple graph having vertex set $V(D)$ and edge set $\{uv \mid (u, w), (v, w) \in A(D) \text{ for some } w \in V(D) \}$. The phylogeny graph of an acyclic digraph $D$, ... More
The Turán number of blow-ups of treesApr 15 2019A conjecture of Erd\H{o}s from 1967 asserts that any graph on $n$ vertices which does not contain a fixed $r$-degenerate bipartite graph $F$ has at most $Cn^{2-1/r}$ edges, where $C$ is a constant depending only on $F$. We show that this bound holds for ... More
From Hall's Marriage Theorem to Boolean Satisfiability and BackApr 15 2019Motivated by the application of Hall's Marriage Theorem in various LP-rounding problems, we introduce a generalization of the classical marriage problem (CMP) that we call the Fractional Marriage Problem. We show that the Fractional Marriage Problem is ... More
A Linear Upper Bound on the Weisfeiler-Leman Dimension of Graphs of Bounded GenusApr 15 2019The Weisfeiler-Leman (WL) dimension of a graph is a measure for the inherent descriptive complexity of the graph. While originally derived from a combinatorial graph isomorphism test called the Weisfeiler-Leman algorithm, the WL dimension can also be ... More
Some cyclic properties of $L_1$-graphsApr 15 2019A graph $G$ is called an $L_1$-graph if $d(u)+d(v)\ge|N(u)\cup N(v)\cup N(w)|-1$ for every triple of vertices $u,v,w$ where $u$ and $v$ are at distance 2 and $w\in N(u)\cap N(v)$. Asratian et al. (1996) proved that all finite connected $L_1$-graphs on ... More
Binary patterns in the Prouhet-Thue-Morse sequenceApr 15 2019We show that, with the exception of the words $a^2ba^2$ and $b^2ab^2$, all (finite or infinite) binary patterns in the Prouhet-Thue-Morse sequence can actually be found in that sequence as segments (up to exchange of letters in the infinite case). We ... More
A decorated tree approach to random permutations in substitution-closed classesApr 15 2019We establish a novel bijective encoding that represents permutations as forests of decorated (or enriched) trees. This allows us to prove local convergence of uniform random permutations from substitution-closed classes satisfying a criticality constraint. ... More
An Asymptotic Form of the Generating Function $\prod_{k=1}^\infty (1+x^k/k)$Apr 15 2019It is shown that the sequence of rational numbers $r(k)$ generated by the ordinary generating function $\prod_{k=1}^\infty (1+x^k/k)$ converges to a limit $C > 0$. $C$ can be expressed as $C = \exp\Bigl(-\sum_{k = 2}^\infty \frac{(-1)^k}{k}\ \zeta(k) ... More
On restricted colorings of $(d,s)$-edge colorable graphsApr 15 2019A cycle is $2$-colored if its edges are properly colored by two distinct colors. A $(d,s)$-edge colorable graph $G$ is a $d$-regular graph that admits a proper $d$-edge coloring in which every edge of $G$ is in at least $s-1$ $2$-colored $4$-cycles. Given ... More
The Varchenko DeterminantApr 15 2019Varchenko introduced a distance function on chambers of a hyperplane arrangement which gives rise to a determinant indexed by chambers whose entry in position $(C,D)$ is the distance between $C$ and $D$, and proved that that determinant has a nice factorization: ... More
Latin cubes of even order with forbidden entriesApr 15 2019We consider the problem of constructing Latin cubes subject to the condition that some symbols may not appear in certain cells. We prove that there is a constant $\gamma > 0$ such that if $n=2t$ and $A$ is a $3$-dimensional $n\times n\times n$ array where ... More
The $h$-edge tolerable diagnosability of balanced hypercubesApr 15 2019To measure the fault diagnosis capability of a multiprocessor system with faulty links, Zhu et al. [Theoret. Comput. Sci. 758 (2019) 1--8] introduced the $h$-edge tolerable diagnosability. This kind of diagnosability is a generalization of the concept ... More
Making multigraphs simple by a sequence of double edge swapsApr 15 2019A double edge swap is an operation on (undirected) loopy multigraphs (multiple edges and multiple loops are allowed) that replaces two edges $(v_1,v_2)$ and $(v_3,v_4)$ by $(v_2,v_3)$ and $(v_4,v_1)$. The swap is admissible if $(v_1,v_2)$ and $(v_3,v_4)$ ... More
Deza graphs with parameters (v,k,k-2,a)Apr 15 2019A Deza graph with parameters $(v,k,b,a)$ is a $k$-regular graph on $v$ vertices in which the number of common neighbors of two distinct vertices takes two values $a$ or $b$ ($a\leq b$) and both cases exist. In the previous papers Deza graphs with parameters ... More
Some results on double triangle descendants of $K_5$Apr 15 2019Double triangle expansion is an operation on $4$-regular graphs with at least one triangle which replaces a triangle with two triangles in a particular way. We study the class of graphs which can be obtained by repeated double triangle expansion beginning ... More
Rank $n$ swapping algebra for GrassmannianApr 15 2019The rank $n$ swapping algebra is the Poisson algebra defined on the ordered pairs of points on a circle using the linking numbers, where a subspace of $(\mathbb{K}^n \times \mathbb{K}^{n*})^r/\operatorname{GL}(n,\mathbb{K})$ is its geometric mode. In ... More
Nonsymmetric Macdonald polynomials via integrable vertex modelsApr 15 2019Starting from an integrable rank-$n$ vertex model, we construct an explicit family of partition functions indexed by compositions $\mu = (\mu_1,\dots,\mu_n)$. Using the Yang-Baxter algebra of the model and a certain rotation operation that acts on our ... More
Interesting identities involving weighted representations of integers as sums of arbitrarily many squaresApr 15 2019In this paper, we obtain formulas for the number of representations of positive integers as sums of arbitrarily many squares (and other polygonal numbers) with a certain natural weighting. The resulting weighted sums give Fourier coefficients of weight ... More
Linear algorithms on Steiner domination of treesApr 14 2019A set of vertices $W$ in a connected graph $G$ is called a Steiner dominating set if $W$ is both Steiner and dominating set. The Steiner domination number $\gamma_{st}(G)$ is the minimum cardinality of a Steiner dominating set of $G$. A linear algorithm ... More
Proof of Aharoni Berger ConjectureApr 14 2019We prove the Aharoni Berger Conjecture
Turán numbers of Berge treesApr 14 2019A classical conjecture of Erd\H{o}s and S\'os asks to determine the Tur\'an number of a tree. We consider variants of this problem in the settings of hypergraphs and multi-hypergraphs. In particular, we determine the Tur\'an number of a hypergraph without ... More
Geometrical Realisations of the Simple Permutoassociahedron by Minkowski sumsApr 14 2019This paper offers a geometrical realisation of simple permutoassociahedra, which has significant importance serving as a topological proof of Mac Lane's coherence. We introduce a family of $n$-polytopes, $PA_{n,c}$, obtained by Minkowski sums such that ... More
On bipartite distance-regular Cayley graphs with diameter $3$Apr 14 2019In this paper, we show that every bipartite distance-regular Cayley graph with diameter $3$ can be constructed on the semidirect product of a group and $\mathbb{Z}_{2}$, except possibly for one case.
Spectra of comb graphs with tailsApr 14 2019Given two graphs, a backbone and a finger, a comb product is a new graph obtained by grafting a copy of the finger into each vertex of the backbone. We study the comb graphs in the case when both components are the paths of order $n$ and $k$, respectively, ... More
Proof of the List Coloring Conjecture for line perfect multigraphsApr 14 2019We prove that for a line perfect multigraph the chromatic index is equal to the list chromatic index. This is a generalization of Galvin's result on bipartite multigraphs.
The number of fuzzy subgroups of a finite abelian group of order $p^{n}q^{m}$Apr 14 2019The purpose of this paper is to determine the number of fuzzy subgroups of a finite abelian group of order $p^{n}q^{m}$. As an application of our main result, explicit formulas for the number of fuzzy subgroups of $\mathbb{Z}_{p}^{n}\times\mathbb{Z}_{q}^{m}$ ... More
Flagged $(\mathcal{P},ρ)$-partitionsApr 14 2019We introduce the theory of $(\mathcal{P},\rho)$-partitions, depending on a poset $\mathcal{P}$ and a map $\rho$ from $\mathcal{P}$ to positive integers. The generating function $\mathfrak{F}_{\mathcal{P},\rho}$ of $(\mathcal{P},\rho)$-partitions is a ... More
Partial sums and generating functions of products of Horadam numbers with indices in arithmetic progressionApr 13 2019The sums $\sum_{j = 0}^k {w_{rj + s} u_{mj + n} z^j }$, $\sum_{j = 0}^k {w_{rj + s} v_{mj + n} z^j }$ and $\sum_{j = 0}^k {w_{rj + s} w_{mj + n} z^j }$ are evaluated; where $r$, $s$, $k$, $m$ and $n$ are arbitrary integers, $z$ is arbitrary, $(w_i)$ is ... More
Minimal Separators in GraphsApr 13 2019The Known Menger's theorem states that in a finite graph, the size of a minimum separator set of any pair of vertices is equal to the maximum number of disjoint paths that can be found between these two vertices. In this paper, we study the minimal separators ... More
Frobenius's last proofApr 13 2019Around about 1917, Issai Schur rediscovered the Rogers-Ramanujan identities, and proved a system of polynomial identities that imply them. Schur wrote that Georg Frobenius (his former advisor) had shown him a simple, direct proof of these polynomial identities. ... More
Lucas sequences in $t$-uniform simplicial complexesApr 13 2019We introduce $t$-uniform simplicial complexes and we show that the lengths of spheres in such complexes are the terms of certain Lucas sequences. We find optimal constants for the linear isoperimetric inequality in the hyperbolic case.
On the $g$-extra connectivity of graphsApr 13 2019Connectivity and diagnosability are two important parameters for the fault tolerant of an interconnection network $G$. In 1996, F\`{a}brega and Fiol proposed the $g$-extra connectivity of $G$. A subset of vertices $S$ is said to be a \emph{cutset} if ... More
Arc-transitive bicirculantsApr 13 2019In this paper, we characterise the family of finite arc-transitive bicirculants. We show that every finite arc-transitive bicirculant is a normal $r$-cover of an arc-transitive graph that lies in one of eight infinite families or is one of seven sporadic ... More
A rotation group whose subspace arrangement is not from a real reflection groupApr 12 2019Apr 18 2019We exhibit a family of real rotation groups whose subspace arrangements are not contained in that of any real reflection group, answering a question of Martino and Singh.
A rotation group whose subspace arrangement is not from a reflection groupApr 12 2019We exhibit a family of real rotation groups whose subspace arrangements are not contained in that of any real reflection group, answering a question of Martino and Singh.
Dominator Chromatic Numbers of Orientations of TreesApr 12 2019In this paper we prove that the dominator chromatic number of every oriented tree is invariant under reversal of orientation. In addition to this marquee result, we also prove the exact dominator chromatic number for arborescences and anti-arborescences ... More
Optimal Domination PolynomialsApr 12 2019Let $G$ be a graph on $n$ vertices and $m$ edges and $D(G,x)$ the domination polynomial of $G$. In this paper we completely characterize the values of $n$ and $m$ for which optimal graphs exist for domination polynomials. We also show that there does ... More
Sylvester-Gallai type theorems for quadratic polynomialsApr 12 2019We prove Sylvester-Gallai type theorems for quadratic polynomials. Specifically, we prove that if a finite collection $\mathcal Q$, of irreducible polynomials of degree at most $2$, satisfy that for every two polynomials $Q_1,Q_2\in {\mathcal Q}$ there ... More
The Complete Lattice of Erdős-Menger SeparationsApr 12 2019F. Escalante and T. Gallai studied in the seventies the structure of different kind of separations and cuts between a vertex pair in a (possibly infinite) graph. One of their results is that if there is a finite separation, then the optimal (i.e. minimal ... More
MAT-free reflection arrangementsApr 12 2019We introduce the class of MAT-free hyperplane arrangements which is based on the Multiple Addition Theorem by Abe, Barakat, Cuntz, Hoge, and Terao. We also investigate the closely related class of MAT2-free arrangements based on a recent generalization ... More
Community Detection in the Sparse Hypergraph Stochastic Block ModelApr 11 2019We consider the community detection problem in sparse random hypergraphs. Angelini et al. (2015) conjectured the existence of a sharp threshold on model parameters for community detection in sparse hypergraphs generated by a hypergraph stochastic block ... More
Selfish Mining and Dyck Words in Bitcoin and Ethereum NetworksApr 11 2019The main goal of this article is to present a direct approach for the formula giving the long-term apparent hashrates of Selfish Mining strategies using only elementary probabilities and combinatorics, more precisely, Dyck words. We can avoid computing ... More
Quasi-popular Matchings, Optimality, and Extended FormulationsApr 11 2019Let $G = (A \cup B,E)$ be an instance of the stable marriage problem where every vertex ranks its neighbors in a strict order of preference. A matching $M$ in $G$ is popular if $M$ does not lose a head-to-head election against any matching $N$. That is, ... More