Unifying approaches by amongst others Archimedes, Kepler, Goldberg, Caspar and Klug, Coxeter, and Conway, and extending on a previous formalization of the concept of local symmetry preserving (lsp) operations, we introduce a formal definition of local operations on plane graphs that preserve orientation-preserving symmetries, but not necessarily orientation-reversing symmetries. This operations include

It is shown that for any choice of four different vertices x1,…,x4 in a 2-block G of order p>3, there is a hamiltonian cycle in G2 containing four different edges xiyi of E(G) for certain vertices yi, i=1,2,3,4. This result is best possible.

The total graph of a graph G, denoted by T(G), is defined to be the graph whose vertices are the vertices and edges of G, with two vertices of T(G) adjacent if and only if the corresponding elements of G are adjacent or incident. In this paper, we investigate the existence of Laplacian perfect state transfer and Laplacian pretty good state transfer in the total graph of an r-regular graph, where r≥2

Let Fq denote the finite field with q elements. The Carlitz rank of a permutation polynomial is an important measure of complexity of a polynomial. In this paper we find a sharp lower bound for the weight of any permutation polynomial with Carlitz rank 2, improving the bound found by Gómez-Pérez, Ostafe and Topuzoğlu in that case.

In this paper, we obtain explicit mass formulae for all Euclidean self-orthogonal and self-dual codes of an arbitrary length over finite commutative chain rings of odd characteristic. With the help of these mass formulae, we classify all Euclidean self-orthogonal and self-dual codes of lengths 2,3,4,5 over the chain ring F5[u]∕ and of lengths 2,3,4 over the chain ring F7[u]∕.

We show that every gammoid has special digraph representations, such that a representation of the dual of the gammoid may be easily obtained by reversing all arcs. In an informal sense, the duality notion of a poset applied to the digraph of a special representation of a gammoid commutes with the operation of forming the dual of that gammoid. We use these special representations in order to define

For a graph G and a positive integer k, a k-list assignment of G is a function L on the vertices of G such that for each vertex v∈V(G), |L(v)|≥k. Let s be a nonnegative integer. Then L is a (k,k+s)-list assignment of G if |L(u)∪L(v)|≥k+s for each edge uv. If for each (k,k+s)-list assignment L of G, G admits a proper coloring φ such that φ(v)∈L(v) for each v∈V(G), then we say G is (k,k+s)-choosable

In this paper we introduce modular knight distance on a chessboard and use it to study the geometry of a class of solutions of the (modular) n-queens problem. Such geometry is related to the movement of a knight in classic chess, therefore we benefit from defining modular knight distance prior to the study. The concept of the distance is incorporated to graph theory with the notion of the knight graph

In this paper, we prove that (−∞,0) is a zero-free interval for chromatic polynomials of a family L0 of hypergraphs and (0,1) is a zero-free interval for chromatic polynomials of a subfamily L0′ of L0 of hypergraphs. These results extend known results on zero-free intervals of chromatic polynomials of graphs and hypergraphs.

We prove that for d−1≥3, the graph of any homology (d−1)-sphere Δ with g2(Δ)≥1 and without missing (d−1)-faces is generically redundantly d-rigid. This confirms a conjecture of Nevo and Novinsky.

Let A be a nonempty finite set of k integers. Given a subset B of A, the sum of all elements of B is called the subset sum of B and we denote it by s(B). For a nonnegative integer α (≤k), let Σα(A)≔{s(B):B⊂A,|B|≥α}.Now, let A=(a1,…,a1︸r1copies,a2,…,a2︸r2copies,…,ak,…,ak︸rkcopies) be a finite sequence of integers with k distinct terms, where ri≥1 for i=1,2,…,k. Given a subsequence B of A, the sum of

For coprime integers N,a,b,c, with 0

Given two graphs G and H, we consider the Ramsey-type problem of finding the minimum integer n (denoted by egrk(G:H)) such that n2≥k and for every N≥n, every rainbow G-free k-coloring (using exactly k colors) of the complete graph KN contains a monochromatic copy of H. In this paper, we determine egrk(K3:K1,t) for all integers t≥1 and k≥3 completely. Let S3+ be the unique graph on four vertices consisting

Let G be an edge-colored graph of order n. The minimum color degree of G, denoted by δc(G), is the largest integer k such that for every vertex v, there are at least k distinct colors on edges incident to v. We say that an edge-colored graph is rainbow if all its edges have different colors. In this paper, we consider vertex-disjoint rainbow triangles in edge-colored graphs. Li (2013) showed that if

In this note, we improve on results of Hoppen, Kohayakawa and Lefmann about the maximum number of edge colorings without monochromatic copies of a star of a fixed size that a graph on n vertices may admit. Our results rely on an improved application of an entropy inequality of Shearer.

For a finite abelian group G, the generalized Erdős–Ginzburg–Ziv constant sk(G) is the smallest m such that a sequence of m elements in G always contains a k-element subsequence which sums to zero. If n=exp(G) is the exponent of G, the previously best known bounds for skn(Cnr) were linear in n and r when k≥2. Via a probabilistic argument, we produce the exponential lower bound s2n(Cnr)>n2[1.25+o(1)]rfor

Hakimi, Schmeichel and Thomassen showed in 1979 that every 4-connected triangulation on n vertices has at least n∕log2n hamiltonian cycles, and conjectured that the sharp lower bound is 2(n−2)(n−4). Recently, Brinkmann, Souffriau and Van Cleemput gave an improved lower bound 125(n−2). In this paper we show that every 4-connected triangulation with O(logn) 4-separators has Ω(n2∕log2n) hamiltonian cycles

We characterise digraphs of directed treewidth one in terms of forbidden butterfly minors and in terms of the structure of the cycle hypergraph to a digraph D, i.e. the hypergraph on vertex set V(D) whose hyperedges correspond to the vertex sets of directed cycles in D.

Let π=(d1,d2,…,dn) and π′=(d1′,d2′,…,dn′) be two non-increasing degree sequences. We say π is majorizated by π′, denoted by π≺π′, if and only if ∑i=1ndi=∑i=1ndi′ and ∑i=1jdi≤∑i=1jdi′ for j=1,2,…,n−1. By using majorization, we show that some graphs are determined by their Aα-spectra. Firstly, we show that the lollipop graph is determined by its Aα-spectrum for 0<α<1, which generalizes the result of

For a fixed graph F, the anti-Ramsey number, AR(n,F), is the maximum number of colors in an edge-coloring of Kn which does not contain a rainbow copy of F. In this paper, we determine the exact value of anti-Ramsey numbers of linear forests for sufficiently large n, and show the extremal edge-colored graphs. This answers a question of Fang, Győri, Lu and Xiao.

Fraise and Thomassen (1987) proved that every (k+1)-strong tournament has a hamiltonian cycle which avoids any prescribed set of k arcs. Bang-Jensen, Havet and Yeo showed in Bang-Jensen et al. (2019) that a number of results concerning vertex-connectivity and hamiltonian cycles in tournaments have analogues when we replace vertex connectivity by arc-connectivity and hamiltonian cycles by spanning eulerian

A directed path-factor of a digraph is a spanning subdigraph consisting of a union of vertex-disjoint directed paths in the digraph. In this paper, we give the following result: If D is a digraph of order n≥(2ℓ−1)k−1, and if dD+(u)+dD−(v)≥n−k for every two distinct vertices u and v with (u,v)∉A(D), then D has a directed path-factor with exactly k directed paths of order at least ℓ(≥2). To show this

A pair (T0,T1) of disjoint sets of vertices of a graph G is called a perfect bitrade in G if any ball of radius 1 in G contains exactly one vertex in T0 and T1 or none simultaneously. The volume of a perfect bitrade (T0,T1) is the size of T0. If C0 and C1 are distinct perfect codes with minimum distance 3 in G then (C0∖C1,C1∖C0) is a perfect bitrade. For any q≥3, r≥1 we construct perfect bitrades of

The Motzkin numbers count the number of lattice paths which go from (0,0) to (n,0) using steps (1,1),(1,0) and (1,−1) and never go below the x-axis. Let Mn,k be the number of such paths with exactly k horizontal steps. We investigate the analytic properties of various combinatorial triangles related to the Motzkin triangle [Mn,k]n,k≥0, including their total positivity, the real-rootedness and interlacing

Let r be a positive integer. The Bermond–Thomassen conjecture states that, a digraph of minimum out-degree at least 2r−1 contains r vertex-disjoint directed cycles. A digraph D is called a local tournament if for every vertex x of D, both the out-neighbours and the in-neighbours of x induce tournaments. Note that tournaments form the subclass of local tournaments. In this paper, we verify that the

We prove that if L(G) immerses Kt then L(mG) immerses Kmt, where mG is the graph obtained from G by replacing each edge in G with a parallel edge of multiplicity m. This implies that when G is a simple graph, L(mG) satisfies a conjecture of Abu-Khzam and Langston. We also show that when G is a line graph, G has a Kt-immersion iff G has a Kt-minor whenever t≤4, but this equivalence fails in both directions

A classical problem in enumerative combinatorics is to count the number of derangements, i.e., permutations with no fixed point. In this paper, we study the cyclic derangement polynomials, which count the number of derangements in the wreath product Cr≀Sn. We first establish the relation between the cyclic Eulerian polynomials and the cyclic derangement polynomials. Then we show that the coefficients

Given a digraph D with m arcs, a bijection τ:A(D)→{1,2,…,m} is an antimagic labeling of D if no two vertices in D have the same vertex-sum, where the vertex-sum of a vertex u in D under τ is the sum of labels of all arcs entering u minus the sum of labels of all arcs leaving u. We say (D,τ) is an antimagic orientation of a graph G if D is an orientation of G and τ is an antimagic labeling of D. Motivated

A family of sets is intersecting if any two sets in the family intersect. Given a graph G and an integer r≥1, let I(r)(G) denote the family of independent sets of size r of G. For a vertex v of G, the family of independent sets of size r that contain v is called an r-star. Then G is said to be r-EKR if no intersecting subfamily of I(r)(G) is bigger than the largest r-star. Let n be a positive integer

Recently, linear codes with a few weights have been constructed and extensively studied due to their applications in secret sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, we construct several classes of linear codes with a few weights over Fp, where p is an odd prime. The weight distributions of these constructed codes are also settled by applications

The independence polynomial of a graph is the generating polynomial for the number of independent sets of each cardinality and its roots are called independence roots. We investigate here purely imaginary independence roots. We show that for all k≥4, there are connected graphs with independence number k and purely imaginary independence roots. We also show that every graph is an induced subgraph of

When many colors appear in edge-colored graphs, it is only natural to expect rainbow subgraphs to appear. This anti-Ramsey problem has been studied thoroughly and yet there remain many gaps in the literature. Expanding upon classical and recent results forcing rainbow triangles to appear, we consider similar conditions which force the existence of a properly colored copy of C4.

We construct partition ideals with highly oscillating growth function. Partition ideal X is a set of integer partitions closed under the subpartition relation. Its growth function p(n,X) counts the number of partitions of n that are in X. We present a partition ideal with growth function being infinitely many times both zero and close to the partition function p(n) of all partitions of n. Our main

A fundamental theorem in the study of Dunwoody manifolds is a classification of finite graphs on 2n vertices that satisfy seven conditions (concerning planarity, regularity, and a cyclic automorphism of order n). Its significance is that if the presentation complex of a cyclic presentation is a spine of a 3-manifold then its Whitehead graph satisfies the first five conditions (the remaining conditions

MacDougall’s conjecture states that every regular graph of degree at least 2 has a vertex-magic total labeling (VMTL) with the lone exception of 2K3. Since there is enormous empirical evidence supporting this conjecture, it is reasonable to seek generalizations. Thus we ask the more general question: to what extent does the degree sequence of a graph determine the existence or nonexistence of a VMTL

A Dyck path is a lattice path in the first quadrant of the xy-plane that starts at the origin and ends on the x-axis and has even length. This is composed of the same number of North-East (X) and South-East (Y) steps. A peak and a valley of a Dyck path are the subpaths XY and YX, respectively. A peak is symmetric if the valleys determining the maximal pyramid containing the peak are at the same level

For a digraph G and v∈V(G), let δ+(v) be the number of out-neighbors of v in G. The Caccetta–Häggkvist conjecture states that for all k≥1, if G is a digraph with n=|V(G)| such that δ+(v)≥n∕k for all v∈V(G), then G contains a directed cycle of length at most k. In [2], N. Lichiardopol proved that this conjecture is true for digraphs with independence number equal to two. In this paper, we generalize

According to an old conjecture of Wegner, the piercing number of a set of axis-parallel rectangles in the plane is at most twice the independence number (or matching number) minus 1, that is, τ(F)≤2ν(F)−1. On the other hand, the current best upper bound, due to Corea et al. (2015), is a Ologlogν(F)2 factor away from the current best lower bound. From the other direction, lower bound constructions with

Let AGL(1,Fq) denote the affine linear group of dimension one over the finite field Fq. We determine the Möbius function of the lattice of subgroups of AGL(1,Fq).

The degree of a vertex v in a hypergraph H is the number of vertices in H adjacent to v, and the maximum degree of H is denoted Δ(H). A hypergraph is called simple if every edge has size at least two and every pair of edges has at most one vertex in common. Let χ′(H) denote the chromatic index of a simple hypergraph H. A well-known conjecture of Erdős, Faber and Lovász says that χ′(H)≤|V(H)|. A generalization

Let V be an (n+l)-dimensional vector space over the finite field Fq with l≥n>0 and W be a fixed l-dimensional subspace of V. We say that an m-dimensional subspace U of V is of type (m,k) if dim(U∩W)=k. Denote the set of all subspaces of type (m,k) in V by M(m,k;n+l,n). The collection of all the subspaces of types (m,0) in V, where 0≤m≤n, is the attenuated space. In this paper, we prove the Hilton–Milner

Let X be an infinite, locally finite, connected, quasi-transitive graph without loops or multiple edges. A graph height function on X is a map adapted to the graph structure, assigning to every vertex an integer, called height. Bridges are self-avoiding walks such that heights of interior vertices are bounded by the heights of the start- and end-vertex. The number of self-avoiding walks and the number

A connectivity function on a set E is a function λ:2E→R such that λ(0̸)=0, that λ(X)=λ(E−X) for all X⊆E, and that λ(X∩Y)+λ(X∪Y)≤λ(X)+λ(Y) for all X,Y⊆E. Graphs, matroids and, more generally, polymatroids have associated connectivity functions. In this paper we give a method for identifying when a connectivity function comes from a graph. This method uses no more than a polynomial number of evaluations

A widely investigated subject in combinatorial geometry, originated from Erdős, is the following. Given a point set P of cardinality n in the plane, how can we describe the distribution of the determined distances? This has been generalized in many directions. In this paper we propose the following variants. What is the maximum number of triangles of unit area, maximum area or minimum area, that can

Super-simple designs are used in other constructions, such as coverings, superimposed codes and perfect hash families etc. The existence of super-simple (v,4,λ)-BIBDs has been determined for λ=2,3,4,5,6 and 9. When λ=7, the necessary conditions of such a design are that v≡1,4 (mod 12) and v≥16. In this paper, we show that these necessary conditions are sufficient.

We introduce generalizations of Macdonald polynomials indexed by n-tuples of partitions and characterized by certain orthogonality and triangularity relations. We prove that they can be explicitly given as products of ordinary Macdonald polynomials depending on special alphabets. With this factorization in hand, we establish their most basic properties, such as explicit formulas for their norm-squared

We initiate the study of applying the Combinatorial Nullstellensatz to the DP-coloring of graphs even though, as is well-known, the Alon–Tarsi theorem does not apply to DP-coloring. We define the notion of good covers of prime order which allows us to apply the Combinatorial Nullstellensatz to DP-coloring. We apply these tools to DP-coloring of the cones of certain bipartite graphs and uniquely 3-colorable

A pebble distribution places a nonnegative number of pebbles on the vertices of a graph G. In graph rubbling, the pebbles can be redistributed using pebbling and rubbling moves, typically with the goal of reaching some target pebble distribution. In graph pebbling, only the pebbling move is allowed. The cover pebbling number is the smallest k such that from any initial distribution of k pebbles, it

Correlation-immune (CI) multi-output Boolean functions have the property of keeping the same output distributions when some input variables are fixed. Recently, a new application of CI functions has appeared in the system of resisting side-channel attacks (SCA). In this paper, three new methods are proposed to characterize the tth-order CI multi-output (n-input and m-output) Boolean functions. The

For a prime power q, a very simple algebraic construction is presented for symmetric configurations of type (q2−q−2)q−1. These configurations play an important role in the discussion of the parameter spectrum of configuration types.

In this work, we describe a construction in which we combine together the idea of a bisymmetric matrix and group rings. Applying this construction over the ring F4+uF4 together with the well known extension and neighbour methods, we construct new self-dual codes of length 68. In particular, we find 41 new codes of length 68 that were not known in the literature before.

If k≥1 and G=(V,E) is a finite connected graph, S⊆V is said a distancek-dominating set if every vertex v∈V is within distance k from some vertex of S. The distance k-domination number γwk(G) is the minimum cardinality among all distance k-dominating sets of G. A set S⊆V is a total dominating set if every vertex v∈V satisfies δS(v)≥1 and the total domination number, denoted by γt(G), is the minimum

In this paper, we introduce a new game invariant of graphs, called game connectivity, which is related to the connectivity of graphs. We investigate many fundamental and significant topics of the game connectivity of graphs, and propose many open problems and conjectures.

Let v=2ms+t be a positive integer, where t divides 2ms, and let J be the subgroup of order t of the cyclic group Zv. An integer Heffter array Ht(m,n;s,k) over Zv relative to J is an m×n partially filled array with elements in Zv such that: (a) each row contains s filled cells and each column contains k filled cells; (b) for every x∈Zv∖J, either x or −x appears in the array; (c) the elements in every

Let s and t be positive integers. We use Pt to denote the path with t vertices and K1,s to denote the complete bipartite graph with parts of size 1 and s respectively. The one-subdivision of K1,s is obtained by replacing every edge {u,v} of K1,s by two edges {u,w} and {v,w} with a new vertex w. In this paper, we give a polynomial-time algorithm for the list 3-coloring problem restricted to the class

The generalized wreath product of permutation groups was introduced by Evdokimov and Ponomarenko in order to study the schurity problem for S-rings over cyclic groups. In this paper we construct the generalized wreath product of permutation groups by a method entirely different from Evdokimov and Ponomarenko’s construction. Then we give a necessary and sufficient condition for the wedge product of

We prove that any Cayley graph G with degree d polynomial growth does not satisfy {f(n)}-containment for any f=o(nd−2). This settles the asymptotic behaviour of the firefighter problem on such graphs as it was known that Cnd−2 firefighters are enough, answering and strengthening a conjecture of Develin and Hartke. We also prove that intermediate growth Cayley graphs do not satisfy polynomial containment

Two colourings of a graph are orthogonal if they have the property that when two vertices are coloured with the same colour in one colouring, then those vertices receive distinct colours in the other colouring. In this paper, orthogonal colourings of Cayley graphs are discussed. Firstly, the orthogonal chromatic number of cycle graphs are completely determined. Secondly, the orthogonal chromatic number

Given a t-(v,k,λ) design, D=(X,B), a zero-sum n-flow of D is a map f:B⟶{±1,…,±(n−1)} such that for any point x∈X, the sum of f over all blocks incident with x is zero. For a positive integer k, we find a zero-sum k-flow for an STS(uw) and for an STS(2v+7) for v≡1(mod4), if there are STS(u), STS(w) and STS(v) such that the STS(u) and STS(v) both have a zero-sum k-flow. In 2015, it was conjectured that

We establish that any connected cubic graph of order n>6 has a minimum vertex-edge dominating set of at most 10n31 vertices, thus affirmatively answering the open question posed by Klostermeyer et al. in Discussiones Mathematicae Graph Theory, https://doi.org/10.7151/dmgt.2175. On the other hand, we present an infinite family of cubic graphs whose γve ratio is equal to 27. Finally, we show that the