2n Vertices Research Articles

A remark on the Ramsey number of the hypercube

A well known conjecture of Burr and Erdős asserts that the Ramsey number r(Qn) of the hypercube Qn on 2n vertices is of the order O(2n). In this paper, we show that r(Qn)=O(22n−cn) for a universal constant c>0, improving upon the previous best known bound r(Qn)=O(22n), due to Conlon, Fox and Sudakov.

Asymptotics of the quantization errors for some Markov-type measures with complete overlaps

Let G be a directed graph with 2N vertices 1,2,…,2N. Let T=(Ti,j)(i,j)∈G be a family of contractive similitudes on Rq. We assume that Ti˜,j˜=Ti,j for every (i˜,j˜) in {(i,j),(i,j+N),(i+N,j),(i+N,j+N)}∩G. We denote by K the Mauldin-Williams fractal determined by T. Let χ be a positive probability vector and P a 2N×2N row-stochastic matrix. We denote by ν the Markov-type measure associated with χ and P. Let μ be the image measure of ν under the natural projection, which is supported on K. We consider the following two cases: 1. G has two strongly connected components consisting of N vertices; 2. G is strongly connected. With some assumptions for G and T, for case 1, we determine the quantization dimension for μ in terms of the spectral radius of a related matrix; we prove that the lower quantization coefficient is always positive and establish a necessary and sufficient condition for the upper one to be finite. For case 2, we express the quantization dimension in terms of a pressure-like function and prove that the upper and lower quantization coefficient are always positive and finite.

Combinatorial aspects of sandpile models on wheel and fan graphs

We study combinatorial aspects of the sandpile model on wheel and fan graphs, seeking bijective characterisations of the model’s recurrent configurations on these families. For wheel graphs, we exhibit a bijection between these recurrent configurations and the set of subgraphs of the cycle graph which maps the level of the configuration to the number of edges of the subgraph. This bijection relies on two key ingredients. The first consists in considering a stochastic variant of the standard Abelian sandpile model (ASM), rather than the ASM itself. The second ingredient is a mapping from a given recurrent state to a canonical minimal recurrent state, exploiting similar ideas to previous studies of the ASM on complete bipartite graphs and Ferrers graphs. We also show that on the wheel graph with 2n vertices, the number of recurrent states with level n is given by the first differences of the central Delannoy numbers. Finally, using similar tools, we exhibit a bijection between the set of recurrent configurations of the ASM on fan graphs and the set of subgraphs of the path graph containing the right-most vertex of the path. We show that these sets are also equinumerous with certain lattice paths, which we name Kimberling paths after the author of the corresponding entry in the Online Encyclopedia of Integer Sequences.

On off-diagonal ordered Ramsey numbers of nested matchings

For two graphs G< and H< with linearly ordered vertex sets, the ordered Ramsey numberr<(G<,H<) is the minimum N such that every red-blue coloring of the edges of the ordered complete graph on N vertices contains a red copy of G< or a blue copy of H<.For a positive integer n, a nested matchingNMn< is the ordered graph on 2n vertices with edges {i,2n−i+1} for every i=1,…,n. We improve bounds on the ordered Ramsey numbers r<(NMn<,K3<) obtained by Rohatgi, we disprove his conjecture by showing 4n+1≤r<(NMn<,K3<)≤(3+5)n+1 for every n≥6, and we determine the numbers r<(NMn<,K3<) exactly for n=4,5. As a corollary, this gives stronger lower bounds on the maximum chromatic number of k-queue graphs for every k≥3. We also prove r<(NMm<,Kn<)=Θ(mn) for arbitrary m and n.We expand the classical notion of Ramsey goodness to the ordered case and we attempt to characterize all connected ordered graphs that are n-good for every n∈N. In particular, we discover a new class of ordered trees that are n-good for every n∈N, extending all the previously known examples.

Relations between global forcing number and maximum anti-forcing number of a graph

The global forcing number of a graph G is the minimal cardinality of an edge subset discriminating all perfect matchings of G, denoted by gf(G). For a perfect matching M of G, the minimal cardinality of an edge subset S⊆E(G)∖M such that G−S has a unique perfect matching is called the anti-forcing number of M. The maximum anti-forcing number of G among all perfect matchings is denoted by Af(G). It is known that the maximum anti-forcing number of a hexagonal system equals the famous Fries number.For a bipartite graph G, we show that gf(G)≥Af(G). Next we extend such result to Birkhoff–von Neumann graphs, whose perfect matching polytopes are characterized solely by nonnegativity and degree constraints, by revealing an odd dumbbell of non-bipartite graphs with a unique perfect matching and minimum degree at least two. Finally, we obtain tight upper and lower bounds on gf(G)−Af(G). For a connected bipartite graph G with 2n vertices, 0≤gf(G)−Af(G)≤12(n−1)(n−2). For non-bipartite case, we have −Occ(G)≤gf(G)−Af(G)≤(n−1)(n−2) by introducing a new nonnegative parameter Occ(G).

On the decomposition of circulant graphs using algorithmic approaches

Many structural models in chemistry, biology, computer science, sociology, and operations research can be analyzed using graph theory. Some examples of these structure models are species movement between regions, molecular bonds, shortest spanning trees, development of computer algorithms. This paper introduces the edge decomposition of circulant graphs with 2n vertices by different graph classes. These circulant graphs are denoted as C2n,n, where n is the degree of the graph C2n,n. We propose two algorithmic approaches for constructing edge decomposition of C2n,n. With aid of the proposed algorithmic approaches, we construct paths decompositions, trees decompositions, disjoint unions of cycles and trees decompositions, and complete bipartite graphs decompositions of C2n,n. In the end, a recursive construction of an edge decomposition, based on orthogonal edge decompositions, is proposed. This helps in using the mutually orthogonal decompositions of the circulant graphs C2n,n in several applications, such as the design of experiments and binary codes generation.

Monochromatic paths and cycles in 2-edge-coloured graphs with large minimum degree

AbstractA graph G arrows a graph H if in every 2-edge-colouring of G there exists a monochromatic copy of H. Schelp had the idea that if the complete graph $K_n$ arrows a small graph H, then every ‘dense’ subgraph of $K_n$ also arrows H, and he outlined some problems in this direction. Our main result is in this spirit. We prove that for every sufficiently large n, if $n = 3t+r$ where $r \in \{0,1,2\}$ and G is an n-vertex graph with $\delta(G) \ge (3n-1)/4$ , then for every 2-edge-colouring of G, either there are cycles of every length $\{3, 4, 5, \dots, 2t+r\}$ of the same colour, or there are cycles of every even length $\{4, 6, 8, \dots, 2t+2\}$ of the samecolour.Our result is tight in the sense that no longer cycles (of length $&gt;2t+r$ ) can be guaranteed and the minimum degree condition cannot be reduced. It also implies the conjecture of Schelp that for every sufficiently large n, every $(3t-1)$ -vertex graph G with minimum degree larger than $3|V(G)|/4$ arrows the path $P_{2n}$ with 2n vertices. Moreover, it implies for sufficiently large n the conjecture by Benevides, Łuczak, Scott, Skokan and White that for $n=3t+r$ where $r \in \{0,1,2\}$ and every n-vertex graph G with $\delta(G) \ge 3n/4$ , in each 2-edge-colouring of G there exists a monochromatic cycle of length at least $2t+r$ .

On Proper Edge 3-Colorings of a Cubic Graph

An edge set of a cubic graph is said to be defining if a 3-coloring of it uniquely determines a proper edge 3-coloring of the graph. It is proved that a cubic graph with 3n edges has a defining set of n edges. It is also proved that for a 3-connected plane cubic graph with 3n edges, each face of which has at most d vertices, there exists a defining set of at most $$ n-\frac{n-2d+3}{3d-8} $$ edges. In both cases, an algorithm finding the desired defining set is constructed. For a connected cubic graph G with 3n edges, a series of polynomials over the field 𝔽3 in n+1 variables is defined so that each of them does not vanish identically if and only if there exists a proper edge 3-coloring of G. Finally, it is proved that a cubic multigraph G on 2n vertices has at most 3 · 2n proper edge 3-colorings. This bound is tight. In the case where G has at most one pair of multiple edges, it is proved that G has at most 9 · 2n−2 proper edge 3-colorings.

Boxes, extended boxes and sets of positive upper density in the Euclidean space

AbstractWe prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: 2n vertices of a fixed n-dimensional rectangular box, the same vertices extended with n points completing three-term arithmetic progressions, and the same vertices extended with n points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature.

An efficient shortest path routing on the hypercube with blocking/faulty nodes

SummaryWe investigate fault‐tolerant shortest path problem in the hypercube between two nodes where some nodes are faulty (or blocked) and thus cannot be used in routing. Previously, several similar problems were studied where proposed algorithms are distributed and local‐information‐based, that is, each node in the network knows only its neighbor's status (faulty or not) and they also look for optimal or near‐optimal paths. There have been studies that established some sufficient conditions for these paths to exist. Since these conditions are only sufficient, there could be shortest paths that will be missed by these conditions. We study the problem under the assumption that for two given nodes, a source node s, a target node t, only s requires to have a global information of the network in order to find a shortest path to t, should it exist. A shortest path is defined as the Hamming distance between s and t. This problem can be solved by trivial algorithms. The first is to try all possible paths. In an n‐dimensional hypercube with 2n vertices, this method would cost at least n! time. Another method is to perform a standard shortest path finding algorithm, which would require at least 2n time. A routing algorithm has been previously developed which is efficient in certain situations. However, in the worst case, its running time could be exponential in the hypercube dimension. We propose an efficient algorithm with running time of O(n3m2), polynomial in n, the hypercube dimension, and m, number of blocking nodes. We gain our efficiency by reducing the routing problem to a permutation problem which can be solved using inclusion‐exclusion principle. We finally use dynamic programming technique to optimally count the terms. With the proposed algorithm, not only can we find a shortest path, if such a path does exist, but we can also count all possible shortest paths.

Planar Whitehead graphs with cyclic symmetry arising from the study of Dunwoody manifolds

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 do not necessarily hold). In this paper we observe that this classification relies implicitly on an unstated 8th condition and that this condition is not necessary for such a presentation complex to be the spine of a 3-manifold. We expand the scope of Dunwoody’s classification by classifying all graphs that satisfy the first five conditions.

Highly neighborly centrally symmetric spheres

In 1995, Jockusch constructed an infinite family of centrally symmetric 3-dimensional simplicial spheres that are cs-2-neighborly. Here we generalize his construction and show that for all d≥3 and n≥d+1, there exists a centrally symmetric d-dimensional simplicial sphere with 2n vertices that is cs-⌈d/2⌉-neighborly. This result combined with work of Adin and Stanley completely resolves the upper bound problem for centrally symmetric simplicial spheres.

Long monochromatic paths and cycles in 2-colored bipartite graphs

Gyárfás and Lehel and independently Faudree and Schelp proved that in any 2-coloring of the edges of Kn,n there exists a monochromatic path on at least 2⌈n∕2⌉ vertices, and this is tight. We prove a stability version of this result which holds even if the host graph is not complete; that is, for every γ≫η>0 and n≥n0(γ), if G is a balanced bipartite graph on 2n vertices with minimum degree at least (3∕4+γ)n, then in every 2-coloring of the edges of G, either there exists a monochromatic cycle on at least (1+η)n vertices, or the coloring of G is close to an extremal coloring — in which case G has a monochromatic path on at least 2⌈n∕2⌉ vertices and a monochromatic cycle on at least 2⌊n∕2⌋ vertices. Furthermore, we determine an asymptotically tight bound on the length of a longest monochromatic cycle in a 2-colored balanced bipartite graph on 2n vertices with minimum degree δn for all 0≤δ≤1.

Ordering of the unicyclic signed graphs with perfect matchings by their minimal energies

Let U?2n be the set of unicyclic signed graphs with perfect matchings having 2n vertices, where ? is a signing function from the edge set of the graphs considered to {-1,1}. The increasing order of the signed graphs among U?2n according to their minimal energies is considered. A relationship between the energies of a unicyclic graph and of its signed graphs is derived. A new integral formula for comparing the energies of two signed graph is introduced. In U?2n with n ? 721, the first 18 signed graphs in the increasing order by their minimal energies are obtained.

Partially ordering the class of invertible trees

A tree T is invertible if and only if T has a perfect matching. In Godsil (1985), Godsil considers an invertible tree T and finds that the matrix A(T)−1 has entries in {0,±1} and is the signed adjacency matrix of a graph which contains T. In this paper, we give a new proof of this theorem, which gives rise to a partial ordering relation on the class of all invertible trees on 2n vertices. In particular, we show that given an invertible tree T whose inverse graph has strictly more edges, we can remove an edge from T and add another edge to obtain a non-isomorphic invertible tree T˜ whose median eigenvalue is strictly greater. This extends naturally to a partial ordering. We find the maximal and minimal elements of this poset and explore the implications about the median eigenvalues of invertible trees.

Xor-Magic Graphs

Abstract A connected graph on 2n vertices is defined to be xor-magic if the vertices can be labeled with distinct n-bit binary numbers in such a way that the label at each vertex is equal to the bitwise xor of the labels on the adjacent vertices. We show that there is at least one 3-regular xor-magic graph on 2n vertices for every n ⩾ 2. We classify the 3-regular xor-magic graphs on 8 and 16 vertices, and give multiple examples of 3-regular xor-magic graphs on 32 vertices, including the well-known Dyck graph.

The rainbow connection number of a watermill graph

Let G be a simple, finite and undirected connected graph. An edge-colored graph G is called rainbow connected, if any two vertices in the graph are connected by a path which the edges have distinct colors. Such a path is called rainbow path. An edge-coloring on a graph G is a map . The smallest number k of colors needed in order to make G rainbow connected is called the rainbow connection number of G, denoted by rc(G). The concepts of rainbow connection were introduced by Chartrand et. al. in 2008. Since then many classes of graphs are studied to find its rainbow connection number. The corona product is called a sun graph with 2n vertices. Let Pm be a path with m vertices. The watermill graph is a Cartesian product of graphs and Pm, which is denoted by W M(m, n). In this paper, we determine the rainbow connection number of a watermill graph W M(m, n).

Graded Betti numbers of crown edge ideals

An n-crown on 2n vertices is a graph obtained from complete bipartite graph by removing edges of a perfect matching. Given a finite simple graph G, one can associate a simplicial complex . In this article, we use combinatorial data from the associated simplicial complex of the crown graph and give formulas to find all Betti numbers of edge ideals of .

Commuting decomposition of Kn1,n2,...,nk through realization of the product A(G)A(GP k )

Abstract In this paper, we introduce the notion of perfect matching property for a k-partition of vertex set of given graph. We consider nontrivial graphs G and GP k , the k-complement of graph G with respect to a kpartition of V(G), to prove that A(G)A(GP k ) is realizable as a graph if and only if P satis_es perfect matching property. For A(G)A(GP k ) = A(Γ) for some graph Γ, we obtain graph parameters such as chromatic number, domination number etc., for those graphs and characterization of P is given for which GP k and Γ are isomorphic. Given a 1-factor graph G with 2n vertices, we propose a partition P for which GP k is a graph of rank r and A(G)A(GP k ) is graphical, where n ≤ r ≤ 2n. Motivated by the result of characterizing decomposable Kn,n into commuting perfect matchings [2], we characterize complete k-partite graph Kn1,n2,...,nk which has a commuting decomposition into a perfect matching and its k-complement.

On Betti numbers of edge ideals of crown graphs

An n-crown $$\mathcal {C}_{n,n}$$ on 2n vertices is a graph obtained from complete bipartite graph $${K}_{n,n}$$ by removing edges of a perfect matching. Given a finite simple graph G, one can associate a simplicial complex $$\Delta (G)$$ . In this paper, we use combinatorial data from the associated simplicial complex $$\Delta (\mathcal {C}_{n,n})$$ of the crown graph $$\mathcal {C}_{n,n}$$ and give a formula to find Betti numbers of the form $$\beta _{i,i+1}$$ of edge ideals of $$\mathcal {C}_{n,n}$$ . We also present a formula to find a particular Betti of the edge ideal of a crown graph. We explicitly compute the projective dimension of the edge ideals of crown graphs using domination parameters of the graphs.