Fractional revival occurs between two vertices in a graph if a continuous-time quantum walk unitarily maps the characteristic vector of one vertex to a superposition of the characteristic vectors of the two vertices. This phenomenon is relevant in quantum information in particular for entanglement generation in spin networks. We study fractional revival in graphs whose adjacency matrices belong to

The eigenvalues of the Hamming graph H(n,q) are known to be λi(n,q)=(q−1)n−qi, 0≤i≤n. The characterization of equitable 2-partitions of the Hamming graphs H(n,q) with eigenvalue λ1(n,q) was obtained by Meyerowitz (2003). We study the equitable 2-partitions of H(n,q) with eigenvalue λ2(n,q). We show that these partitions are reduced to equitable 2-partitions of H(3,q) with eigenvalue λ2(3,q) with the

Recently, Bóna and Smith defined strong pattern avoidance, saying that a permutation π strongly avoids a pattern τ if π and π2 both avoid τ. They conjectured that for every positive integer k, there is a permutation in Sk3 that strongly avoids 123⋯(k+1). We use the Robinson–Schensted–Knuth correspondence to settle this conjecture, showing that the number of such permutations is at least kk3∕2+O(k3∕logk)

We consider the following type of question: Given a partial proper d-edge coloring of the d-dimensional hypercube Qd, and lists of allowed colors for the non-colored edges of Qd, can we extend the partial coloring to a proper d-edge coloring using only colors from the lists? We prove that this question has a positive answer in the case when both the partial coloring and the color lists satisfy certain

Let G be a k-uniform hypergraph, and LG be its Laplacian tensor. The β(G) denotes the maximum number of linearly independent nonnegative eigenvectors of LG corresponding to the eigenvalue 0. In this paper, β(G) is called the geometry connectivity of G. We show that the number of connected components of G equals the geometry connectivity β(G).

In this note we obtain a new bound for the acyclic edge chromatic number a′(G) of a graph G with maximum degree Δ proving that a′(G)≤3.569(Δ−1). To get this result we revisit and slightly modify the method described in Kirousis et al. (2017).

Let Knr;λ1,λ2 be the r-partite multigraph in which each part has size n, where two vertices in the same part or different parts are joined by exactly λ1 edges or λ2 edges, respectively. It is proved that there exists a maximal set of t edge-disjoint Hamilton cycles in Knr;λ1,λ2 for λ2n⌊r+34⌋≤t≤min{⌊λ2n2(r−1)2⌋,⌊λ1(n−1)+λ2n(r−1)2⌋}, the upper bound being best possible. The results proved make use of

A weak order is a way n competitors can rank in an event, where ties are allowed. A weak order can also be thought of as a relation on {1,2,…,n} that is transitive and complete. We provide the first efficient algorithms to construct universal cycles for weak orders by considering both rank and height representations. Each algorithm constructs the universal cycles in O(n) time per symbol using O(n)

Consider a matrix M chosen uniformly at random from a class of m×n matrices of zeros and ones with prescribed row and column sums. A partially filled matrix D is a defining set for M if M is the unique member of its class that contains the entries in D. The size of a defining set is the number of filled entries. A critical set is a defining set for which the removal of any entry stops it being a defining

We consider the case in which mixed graphs (with both directed and undirected edges) are Cayley graphs of Abelian groups. In this case, some Moore bounds were derived for the maximum number of vertices that such graphs can attain. We first show these bounds can be improved if we know more details about the order of some elements of the generating set. Based on these improvements, we present some new

A weak order is a way to rank n objects where ties are allowed. In this paper, we develop a generic framework to generate Gray codes for weak orders. We then describe a simple algorithm based on the framework that generates cyclic 2-Gray codes for weak orders in constant amortized time per string. This is the first known cyclic 2-Gray codes for weak orders. The framework can easily be modified to generate

In this paper we introduce and study new notions of uniform recurrence in multidimensional words. A d-dimensional word is called uniformly recurrent if for all (s1,…,sd)∈Nd there exists n∈N such that each block of size (n,…,n) contains the prefix of size (s1,…,sd). We are interested in a modification of this property. Namely, we ask that for each rational direction (q1,…,qd), each rectangular prefix

In this paper we extend the notion of digraphical regular representations in the context of Haar digraphs. Given a group G, a Haar digraph Γ over G is a bipartite digraph having a bipartition {X,Y} such that G is a group of automorphisms of Γ acting regularly on X and on Y. We say that G admits a Haar digraphical representation (HDR for short), if there exists a Haar digraph over G such that its automorphism

Let P(3)(2,4) be the hypergraph having vertex set {1,2,3,4} and edge set {{1,2,3},{1,2,4}}. In this paper we consider vertex colorings of P(3)(2,4)-designs in such a way any block is neither monochromatic nor polychromatic. We find bounds for the upper and lower chromatic numbers, showing also that these bounds are sharp. Indeed, for any admissible v there exists a P(3)(2,4)-design of order v having

We study several enumeration problems connected to linear trees, a broad class which includes stars, paths, generalized stars, and caterpillars. We provide generating functions for counting the number of linear trees on n vertices, characterize the asymptotic growth rate of the number of nonisomorphic linear trees, and show that the distribution of k-linear trees on n vertices follows a central limit

It is shown that the graph obtained by merging two vertices of two 4-cycles is not a Θ-graceful partial cube, thus answering in the negative a question by Brešar and Klavžar from Brešar and Klavžar (2006), who asked whether every partial cube is Θ-graceful.

For a finite point set in Rd, we consider a peeling process where the vertices of the convex hull are removed at each step. The layer number L(X) of a given point set X is defined as the number of steps of the peeling process in order to delete all points in X. It is known that if X is a set of random points in Rd, then the expectation of L(X) is Θ(|X|2∕(d+1)), and recently it was shown that if X is

Let k≥6. We prove that any large enough finite group G contains k elements which span quadratically many triples of the form (a,b,ab)∈S×G, given any dense set S⊆G×G. The quadratic bound is asymptotically optimal. In particular, this provides an elementary proof of a conjecture of Brown, Erdős and Sós, in finite groups and when k is large.

Given a bipartite graph G=(V,E) with bipartition V=A∪B, where |A|=|B|. We present a new construction of pairs of cospectral bipartite graphs by “rotating” G in two different ways. More precisely, let l and k be two positive integers with l≠k. Take l copies of A (resp. B) and k copies of B (resp. A), say A1,A2,…,Al and B1,B2,…,Bk (resp. B1′,B2′,…,Bl′ and A1′,A2′,…,Ak′). Let Γ (resp. Γ′) be the graph

We study the abstract regular polyhedra with automorphism groups that act faithfully on their vertices, and show that each non-flat abstract regular polyhedron covers a “vertex-faithful” polyhedron with the same number of vertices. We then use this result and earlier work on flat polyhedra to study abstract regular polyhedra based on the size of their vertex set. In particular, we classify all regular

In this paper, we study three designs related to the graph K4−e. We complete the determination of the spectrum of uniform frames, maximum resolvable packings and minimum resolvable coverings of K4−e.

An ordered graph is a graph equipped with a linear ordering of its vertex set. A pair of ordered graphs is Ramsey finite if it has only finitely many minimal ordered Ramsey graphs and Ramsey infinite otherwise. Here an ordered graph F is an ordered Ramsey graph of a pair (H,H′) of ordered graphs if for any coloring of the edges of F in colors red and blue there is either a copy of H with all edges

A Gallai coloring of a complete graph Kn is an edge coloring without triangles colored with three different colors. A sequence e1≥⋯≥ek of positive integers is an (n,k)-sequence if ∑i=1kei=n2. An (n,k)-sequence is a G-sequence if there is a Gallai coloring of Kn with k colors such that there are ei edges of color i for all i,1≤i≤k. Gyárfás, Pálvölgyi, Patkós and Wales proved that for any integer k≥3

It is well known that the circular flow number of a bridgeless cubic graph can be computed in terms of certain partitions of its vertex set with prescribed properties. In the present paper, we first study some of these properties that turn out to be useful in order to make an efficient and practical implementation of an algorithm for the computation of the circular flow number of a bridgeless cubic

To determine the Turán numbers of even cycles is a central problem of extremal graph theory. Even for C6, the Turán number is still open. Till now, the best known upper bound is given by Füredi, Naor and Verstraëte [On the Turán number for the hexagon, Advances in Math.]. In 2010, Nikiforov posed a spectral version of extremal graph theory problem: what is the maximum spectral radius ρ of an H-free

In this note we show every orientation of a connected cubic graph admits an oriented 8-colouring. This lowers the best-known upper bound for the chromatic number of the family of orientations of connected cubic graphs. We further show that every such oriented graph admits a 2-dipath 7-colouring. These results imply that either the oriented chromatic number for the family of orientations of connected

Let f be a binary word and n≥1. Then the generalized Lucas cube Qn(f↽) is the graph obtained from the n-cube Qn by removing all vertices that have a circulation containing f as a factor. Ilić, Klavžar and Rho solved the question for which f and n, Qn(f↽) is an isometric subgraph of Qn for all binary words of length at most five. This question is further studied in this paper. For an isometric word

A long standing open problem in extremal graph theory is to describe all graphs that maximize the number of induced copies of a path on four vertices. The character of the problem changes in the setting of oriented graphs, and becomes more tractable. Here we resolve this problem in the setting of oriented graphs without transitive triangles.

A family of sets is called r-cover free if no set in the family is contained in the union of r (or less) other sets in the family. A 1-cover free family is simply an antichain with respect to set inclusion. Thus, Sperner’s classical result determines the maximal cardinality of a 1-cover free family of subsets of an n-element set. Estimating the maximal cardinality of an r-cover free family of subsets

Let U be a (k−12−1)-dimensional subspace of quadratic forms defined on Fk with the property that U does not contain any reducible quadratic form. Let V(U) be the points of PG(k−1,F) which are zeros of all quadratic forms in U. We will prove that if there is a group G which fixes U and no line of PG(k−1,F) and V(U) spans PG(k−1,F) then any hyperplane of PG(k−1,F) is incident with at most k points of

A fractional matching of a graph G is a function f giving each edge a number in [0,1] so that ∑e∈Γ(v)f(e)≤1 for each vertex v∈V(G), where Γ(v) is the set of edges incident to v. The fractional matching number of G, written α∗′(G), is the maximum value of ∑e∈E(G)f(e) over all fractional matchings. In this paper, we investigate the relations between the fractional matching number and the signless Laplacian

A (proper) total-k-coloring ϕ:V(G)∪E(G)→{1,2,…,k} is called adjacent vertex distinguishing if Cϕ(u)≠Cϕ(v) for each edge uv∈E(G), where Cϕ(u) is the set of the color of u and the colors of all edges incident with u. We use χa′′(G) to denote the smallest value k in such a coloring of G. Zhang et al. (2005) first introduced this coloring and conjectured that χa′′(G)≤Δ(G)+3 for every simple graph G. It

In this article we extend a theorem of Andrews, Crippa, and Simon on the asymptotic behavior of polynomials defined by a general class of recursive equations. Here the polynomials are in the variable q, and the recursive definition at step n introduces a polynomial in n. Our extension replaces the polynomial in n with either an exponential or periodic function of n.

Let T be a non-star tree of order n and maximum degree Δ. Hong Wang showed that there is a packing σ such that the degree of every vertex v in T⊕σ(T) does not exceed Δ(T)+2. In this note we improve this result in two directions. First, we show that with only one exception, there exists a packing σ such that Δ(T⊕σ(T))≤Δ(T)+1. Moreover, this bound is, in a sense, uniform.

Given an arbitrary sequence of non-negative integers λ→=(λ1,…,λn) and a graph G with vertex set {v1,…,vn}, the bounded degree complex, denoted as BDλ→(G), is a simplicial complex whose faces are the subsets H⊆E(G) such that for each i∈{1,…,n}, the degree of vertex vi in the induced subgraph G[H] is at most λi. When λi=k for all i, the bounded degree complex BDλ→(G) is called the k-matching complex

For two s-uniform hypergraphs H and F, the Turán number exs(H,F) is the maximum number of edges in an F-free subgraph of H. Let s,r,k,n1,…,nr be integers satisfying 2≤s≤r and n1≤n2≤⋯≤nr. De Silva, Heysse and Young determined ex2(Kn1,…,nr,kK2) and De Silva, Heysse, Kapilow, Schenfisch and Young determined ex2(Kn1,…,nr,kKr). In this paper, as a generalization of these results, we consider three Turán-type

Given a central integral arrangement, the reduction of the arrangement modulo a positive integer q gives rise to a subgroup arrangement in Zqℓ. Kamiya et al. (2008) introduced the notion of characteristic quasi-polynomial, which enumerates the cardinality of the complement of this subgroup arrangement. Chen and Wang (2012) found a similar but more general setting that replacing the integral arrangement

Given a signed graph Σ with n vertices, let μ be an eigenvalue of Σ, and let t be the codimension of the corresponding eigenspace. We prove that n≤t+23whenever μ∉{0,1,−1}. We show that this bound is sharp by providing examples of signed graphs in which it is attained. We also discuss particular cases in which the bound can be decreased.

Lovász (1965) characterized graphs without two vertex-disjoint cycles, which implies that such graphs have at most three vertices hitting all cycles. In this paper, we ask whether such a small hitting set exists for S-cycles, when a graph has no two vertex-disjoint S-cycles. For a graph G and a vertex set S of G, an S-cycle is a cycle containing a vertex of S. We provide an example G on 21 vertices

Let G be a graph, and let χG be its chromatic polynomial. For any non-negative integers i,j, we give an interpretation for the evaluation χG(i)(−j) in terms of acyclic orientations. This recovers the classical interpretations due to Stanley and to Greene and Zaslavsky respectively in the cases i=0 and j=0. We also give symmetric function refinements of our interpretations, and some extensions. The

The core problem of discrete tomography, i.e., the faithful reconstruction of an unknown discrete object from projections along a given set of directions, is ill-posed in general. When further constraints are imposed on the object or on the employed directions, uniqueness of the reconstruction can be obtained. It is the case, for example, of convex lattice sets in Z2, for which a theorem by Gardner

Laplacian eigenvalues of trees have been studied by many researchers. They are closely related to the diameter, domination number, matching number and Laplacian energy of trees. The goal of this paper is to prove a conjecture on the number of Laplacian eigenvalues of trees that are less than the average degree.

Let T be a tree with n vertices, and Dn be the distance matrix of T. Graham and Pollak (1971) discovered an elegant formula for the determinant of Dn: det(Dn)=−(n−1)(−2)n−2. It is surprising that it depends only on the order of T, not on the specific structure of T. By virtue of the classical Dodgson’s determinant-evaluation rule, Yan and Yeh (2006) presented a simple proof of the formula above. In

We determine the number of Fq-rational points of hyperplane sections of classical determinantal varieties defined by the vanishing of minors of a fixed size of a generic matrix, and identify the hyperplane sections giving the maximum number of Fq-rational points. Further we consider similar questions for sections by linear subvarieties of a fixed codimension in the ambient projective space. This is

Let c and c′ be edge or vertex colourings of a graph G. The stabiliser of c is the set of automorphisms of G that preserve the colouring. We say that c′ is less symmetric than c if the stabiliser of c′ is contained in the stabiliser of c. We show that if G is not a bicentred tree, then for every vertex colouring of G there is a less symmetric edge colouring with the same number of colours. On the other

Let D be 2-(v,k,λ) symmetric designs, let G≤Aut(D) be flag-transitive. In this paper, all symmetric designs D with λ≥(k,λ)2 and Soc(G)=An for n≥5 are classified.

We provide generating functions for the popularity and the distribution of patterns of length at most three over the set of Dyck paths having a first return decomposition constrained by height.

Simple nontrivial s-resolvable t-designs for t≥3 and 1

Left and right idealizers are important invariants of linear rank-distance codes. In the case of maximum rank-distance (MRD for short) codes in Fqn×n the idealizers have been proved to be isomorphic to finite fields of size at most qn. Up to now, the only known MRD codes with maximum left and right idealizers are generalized Gabidulin codes, which were first constructed in 1978 by Delsarte and later

Let G be a connected non-odd-bipartite hypergraph with even uniformity. The least H-eigenvalue of the signless Laplacian tensor of G is simply called the least eigenvalue of G and the corresponding H-eigenvectors are called the first eigenvectors of G. In this paper we give some numerical and structural properties about the first eigenvectors of G which contains an odd-bipartite branch, and investigate

A vertex coloring of a graph G is called distinguishing (or symmetry breaking) if no non-identity automorphism of G preserves it, and the distinguishing number, shown by D(G), is the smallest number of colors required for such a coloring. This paper is about counting non-equivalent distinguishing colorings of graphs with k colors. A parameter, namely Φk(G), which is the number of non-equivalent distinguishing

For a fixed set of positive integers R, we say H is an R-uniform hypergraph, or R-graph, if the cardinality of each edge belongs to R. An R-graph H is covering if every vertex pair of H is contained in some hyperedge. For a graph G=(V,E), a hypergraph H is called a Berge-G, denoted by BG, if there is an injection i:V(G)→V(H) and a bijection f:E(G)→E(H) such that for all e=uv∈E(G), we have {i(u),i(v)}⊆f(e)

Erdős, Hajnal and Rado asked whether ℵω1ℵ2→ℵω1ℵ02 and whether ℵω1ℵ2→ℵω1ℵ12. We shall prove that both relations are independent over ZFC. We shall also prove that μℵ2→μℵ22 is independent over ZF for some μ>cf(μ)=ω1.

We introduce the Maker–Breaker domination game, a two player game on a graph. At his turn, the first player, Dominator, selects a vertex in order to dominate the graph while the other player, Staller, forbids a vertex to Dominator in order to prevent him to reach his goal. Both players play alternately without missing their turn. This game is a particular instance of the so-called Maker–Breaker games

In Goppa (1970), V. D. Goppa presents a new family of linear codes over finite fields. This family, now known as Goppa codes, has been studied for nearly 50 years, including efficient decoding algorithms (Patterson, 1975) and its applications in cryptography (McEliece, 1978). Moreover, in Andrade et al. (2005), de Andrade and Palazzo present a generalization of Goppa codes to finite rings. Motivated

A point set in a metric space is said to have property Φ(4,5) if every 4 elements determine at least 5 distinct distances. According to an old conjecture of Erdős (1986 or earlier), a set of n points in the Euclidean plane satisfying this restriction determines Ω(n2) distinct distances. This property (restriction) is shown to be equivalent to forbidding eight 4-element patterns, πi, i=1,…,8 (described

For a given set S⊆Zm and n¯∈Zm, RS(n¯) is defined as the number of solutions of the equation n¯=s¯+s′¯ with unordered pair (s¯,s′¯)∈S2 and s¯≠s′¯. In this paper, we prove that if m is an even positive integer and not a power of 2 then there exist two distinct sets A,B with A∪B=Zm, |A∩B|=2, B≠A+m∕2¯ such that RA(n¯)=RB(n¯) for all n¯∈Zm. If m is a power of 2, A∪B=Zm, |A∩B|=2, then RA(n¯)=RB(n¯) for

Let st={st1,…,stk} be a set of k statistics on permutations with k≥1. We say that two given subsets of permutations T and T′ are st-Wilf-equivalent if the joint distributions of all statistics in st over the sets of T-avoiding permutations Sn(T) and T′-avoiding permutations Sn(T′) are the same. The main purpose of this paper is the (cr,nes)-Wilf-equivalence classes for all single patterns in S3, where

In this paper, we introduce Eulerian and even-face ribbon graph minors. These minors preserve Eulerian and even-face properties of ribbon graphs, respectively. We then characterize Eulerian, even-face, plane Eulerian and plane even-face ribbon graphs using these minors.

The article gives a combinatorial characterisation of a ruled cubic surface in PG(4,q). In PG(4,q), q odd, q≥9, a set K of q2+2q+1 points is a ruled cubic surface if and only if K has class [q+1,2q+1,3q+1]3 such that every plane that contains four points of K contains at least q+1 points of K.