Extended skolem sequences

A k-extended Skolem sequence of order n is an integer sequence (s₁, s₂,…, s_2n+1) in which s_k = 0 and for each j ? {1,…,n}, there exists a unique i ? {1,…, 2n} such that s_i = s_i+j = j. We show that such a sequence exists if and only if either 1) k is odd and n ≡ 0 or 1 (mod 4) or (2) k is even and n ≡ 2 or 3 (mod 4). The same conditions are also shown to be necessary and sufficient for the existence of excess Skolem sequences. Finally, we use extended Skolem sequences to construct maximal cyclic partial triple systems. © 1995 John Wiley & Sons, Inc.     

Isomorphism of circulant graphs and digraphs

Let S? {1, …, n?1} satisfy ?S = S mod n. The circulant graph G(n, S) with vertex set {v₀, v₁,…, v_n?1} and edge set E satisfies v_iv_j?E if and only if j ? i ∈ S, where all arithmetic is done mod n. The circulant digraph G(n, S) is defined similarly without the restriction S = ? S. Ádám conjectured that $\text{G(n, S) ? G(n, S′)}$ if and only if S = uS′ for some unit u mod n. In this paper we prove the conjecture true if n = pq where p and q are distinct primes. We also show that it is not generally true when n = p², and determine exact conditions on S that it be true in this case. We then show as a simple consequence that the conjecture is false in most cases when n is divisible by p² where p is an odd prime, or n is divisible by 2⁴.     

Utilizing Shelve Slots: Sufficiency Conditions for Some Easy Instances of Hard Problems

In this paper we present a minimal set of conditions sufficient to assure the existence of a solution to a system of nonnegative linear diophantine equations. More specifically, suppose we are given a finite item set U = {u₁, u₂, . . . , u_k} together with a "size" v_i ≡ v(u_i) ∈ Z⁺, such that v_i ≠ v_j for i ≠ j, a "frequency" a_i ≡ a(u_i) ∈ Z⁺, and a positive integer (shelf length) L ∈ Z⁺ with the following conditions: (i) L = ∏ⁿ_j=1p_j(p_j ∈ Z⁺ ∀j, p_j ≠ p_l for j ≠ l) and v_i = ∏ _j∈Aip_j, A_i ⊆ {l, 2, . . . , n} for i = 1, . . . , n; (ii) (A_i\{⋂^k_j=1A_j}) ∩ (A_l\{⋂^k_j=1A_j}) = ⊘∀i ≠ l. Note that v_i|L (divides L) for each i. If for a given m ∈ Z⁺, ∑ⁿ_i=1a_iv_i = mL (i.e., the total size of all the items equals the total length of the shelf space), we prove that conditions (i) and (ii) are sufficient conditions for the existence of a set of integers {b₁₁, b₁₂, . . . , b_1m, b₂₁, . . . , b_n1, . . . , b_nm}⊆ N such that ∑^m_j=1b_ij = a_i, i = 1, . . . , k, and ∑^k_i=1b_ijv_i = L, j =1, . . . , m (i.e., m shelves of length L can be fully utilized). We indicate a number of special cases of well known NP-complete problems which are subsequently decided in polynomial time.     

Isomorphism Checking of I-graphs

We consider the class of I-graphs, which is a generalization of the class of the generalized Petersen graphs. We show that two I-graphs I(n, j, k) and I(n, j ₁, k ₁) are isomorphic if and only if there exists an integer a relatively prime to n such that either {j ₁, k ₁} =? {a j mod n, a k mod n } or {j ₁, k ₁} =? {a j mod n, ? a k mod n }. This result has an application in the enumeration of non-isomorphic I-graphs and unit-distance representations of generalized Petersen graphs.     

Existence of frame sols of type 2nu1

An SOLS (self-orthogonal latin square) of order n with n₁ missing sub-SOLS (holes) of order h_i (1 ? i ? k), which are disjoint and spanning (i.e., Σ _1?i?kn_ih_i = n), is called a frame SOLS and denoted by FSOLS(h₁ⁿ¹ h₂ⁿ² …h_k^nk). In this article, it is shown that for u ? 2, an FSOLS(2ⁿu¹) exists if and only if n ? 1 + u. © 1995 John Wiley & Sons, Inc.     

两类广义Petersen 图的Euler亏格

广义 Petersen 图 P(n, m) 是这样的一个图:它的顶点集是{u_i, v_i | i=0,1, ^… , n-1}, 边集是 {u_iu_i+1, v_iv_i+m, u_iv_i | i=0,1, ^…, n-1}, 这里 m, n 是正整数、加法是在模n 下且 m<|n/2| . 这篇文章证明了P(2m+1, m)(m≥ 2) 的 Euler 亏格是1, 并且 P(2m+2, m)(m≥ 5) 的 Euler 亏格是2.     

GI-graphs: a new class of graphs with many symmetries

The class of generalized Petersen graphs was introduced by Coxeter in the 1950s. Frucht, Graver and Watkins determined the automorphism groups of generalized Petersen graphs in 1971, and much later, Nedela and ?koviera and (independently) Lovre?i?-Sara?in characterised those which are Cayley graphs. In this paper we extend the class of generalized Petersen graphs to a class of GI-graphs. For any positive integer n and any sequence j ₀,j ₁,…,j _t?1 of integers mod n, the GI-graph GI(n;j ₀,j ₁,…,j _t?1) is a (t+1)-valent graph on the vertex set \(\mathbb{Z}_{t} \times\mathbb{Z}_{n}\) , with edges of two kinds:

• an edge from (s,v) to (s′,v), for all distinct \(s,s' \in \mathbb{Z}_{t}\) and all \(v \in\mathbb{Z}_{n}\) ,
• edges from (s,v) to (s,v+j _s ) and (s,v?j _s ), for all \(s \in\mathbb{Z}_{t}\) and \(v \in\mathbb{Z}_{n}\) .

By classifying different kinds of automorphisms, we describe the automorphism group of each GI-graph, and determine which GI-graphs are vertex-transitive and which are Cayley graphs. A GI-graph can be edge-transitive only when t≤3, or equivalently, for valence at most 4. We present a unit-distance drawing of a remarkable GI(7;1,2,3).     

Z-cyclic triplewhist tournaments when the number of players involves primes of the form 8u + 5

When the number of players, v, in a whist tournament, Wh(v), is ≡ 1 (mod 4) the only instances of a Z-cyclic triplewhist tournament, TWh(v), that appear in the literature are for v = 21,29,37. In this study we present Z-cyclic TWh(v) for all v ∈ T = {v = 8u + 5: v is prime, 3 ≤ u ≤ 249}. Additionally, we establish (1) for all v ∈ T there exists a Z-cyclic TWh(vⁿ) for all n ≥ 1, and (2) if v_i ∈ T, i = 1,…,n, there exists a Z-cyclic TWh(v… v) for all ?_i ≥ 1. It is believed that these are the first instances of infinite classes of Z-cyclic TWh(v), v ≡ 1 (mod 4). © 1994 John Wiley & Sons, Inc.     

Mutually Independent Hamiltonian Connectivity of (n, k)-Star Graphs

A graph G is hamiltonian connected if there exists a hamiltonian path joining any two distinct nodes of G. Two hamiltonian paths and of G from u to v are independent if u = u ₁ = v ₁, v = u _v(G) = v _v(G) , and u _i ≠ v _i for every 1 < i < v(G). A set of hamiltonian paths, {P ₁, P ₂, . . . , P _k }, of G from u to v are mutually independent if any two different hamiltonian paths are independent from u to v. A graph is k mutually independent hamiltonian connected if for any two distinct nodes u and v, there are k mutually independent hamiltonian paths from u to v. The mutually independent hamiltonian connectivity of a graph G, IHP(G), is the maximum integer k such that G is k mutually independent hamiltonian connected. Let n and k be any two distinct positive integers with n–k ≥ 2. We use S _n,k to denote the (n, k)-star graph. In this paper, we prove that IHP(S _n,k ) = n–2 except for S _4,2 such that IHP(S _4,2) = 1.      

Cleaning Random d-Regular Graphs with Brooms

A model for cleaning a graph with brushes was recently introduced. Let α = (v ₁, v ₂, . . . , v _n ) be a permutation of the vertices of G; for each vertex v _i let ${N^+(v_i)=\{j: v_j v_i \in E {\rm and} j>\,i\}}${N^+(v_i)=\{j: v_j v_i \in E {\rm and} j>\,i\}} and N^-(v_i)={j: v_j v_i ? E and j <  i}{N^-(v_i)=\{j: v_j v_i \in E {\rm and} j<\,i\}} ; finally let b_a(G)=?_i=1ⁿ max{|N⁺(v_i)|-|N^-(v_i)|,0}{b_{\alpha}(G)=\sum_{i=1}^n {\rm max}\{|N^+(v_i)|-|N^-(v_i)|,0\}}. The Broom number is given by B(G) =  max _α b _α (G). We consider the Broom number of d-regular graphs, focusing on the asymptotic number for random d-regular graphs. Various lower and upper bounds are proposed. To get an asymptotically almost sure lower bound we use a degree-greedy algorithm to clean a random d-regular graph on n vertices (with dn even) and analyze it using the differential equations method (for fixed d). We further show that for any d-regular graph on n vertices there is a cleaning sequence such at least n(d + 1)/4 brushes are needed to clean a graph using this sequence. For an asymptotically almost sure upper bound, the pairing model is used to show that at most n(d+2?{d ln2})/4{n(d+2\sqrt{d \ln 2})/4} brushes can be used when a random d-regular graph is cleaned. This implies that for fixed large d, the Broom number of a random d-regular graph on n vertices is asymptotically almost surely \fracn4(d+Q(?d)){\frac{n}{4}(d+\Theta(\sqrt{d}))}.     

Sphere and Dot Product Representations of Graphs

A graph G is a k-sphere graph if there are k-dimensional real vectors v ₁,…,v _n such that ij∈E(G) if and only if the distance between v _i and v _j is at most 1. A graph G is a k-dot product graph if there are k-dimensional real vectors v ₁,…,v _n such that ij∈E(G) if and only if the dot product of v _i and v _j is at least 1.     

Asymptotic behaviour for a nonlocal diffusion equation on a lattice

In this paper we study the asymptotic behaviour as t → ∞ of solutions to a nonlocal diffusion problem on a lattice, namely, u^￠_n(t) = ?_{j ? \mathbbZ^d} J_n-ju_j(t)-u_n(t)u^{\prime}_{n}(t) = \sum_{{j\in}{{{\mathbb{Z}}}^{d}}} J_{n-j}u_{j}(t)-u_{n}(t) with t ≥ 0 and n ? \mathbbZ^dn \in {\mathbb{Z}}^{d}. We assume that J is nonnegative and verifies ?_{n ? \mathbbZ^d}J_n = 1\sum_{{n \in {\mathbb{Z}}}^{d}}J_{n}= 1. We find that solutions decay to zero as t → ∞ and prove an optimal decay rate using, as our main tool, the discrete Fourier transform.     

Note on the local growth of iterated polynomials

The local behavior of the iterates of a real polynomial is investigated. The fundamental result may be stated as follows: THEOREM. Let x_i, for i=1, 2, ..., n+2, be defined recursively by x_i+1=f(x_i), where x₁ is an arbitrary real number and f is a polynomial of degree n. Let x_i+1?x_i≧1 for i=1, ..., n + 1. Then for all i, 1 ≦i≦n, and all k, 1≦k≦n+1?i, $$ - \frac{{2^{k - 1} }}{{k!}}< f\left[ {x_1 ,... + x_{i + k} } \right]< \frac{{x_{i + k + 1} - x_{i + k} + 2^{k - 1} }}{{k!}},$$ where f[x_i, ..., x_i+k] denotes the Newton difference quotient. As a consequence of this theorem, the authors obtain information on the local behavior of the solutions of certain nonlinear difference equations. There are several cases, of which the following is typical: THEOREM. Let {x_i}, i = 1, 2, 3, ..., be the solution of the nonlinear first order difference equation x_i+1=f(x_i) where x₁ is an arbitrarily assigned real number and f is the polynomial \(f(x) = \sum\limits_{j = 0}^n {a_j x^j } ,n \geqq 2\) . Let δ be positive with δ^n?1=|2^n?1/n!a_n|. Then, if n is even and a_n<0, there do not exist n + 1 consecutive increments Δx_i=x_i+1?x_i in the solution {x_i} with Δx_i≧δ. The special case in which the iterated polynomial has integer coefficients leads to a “nice” upper bound on a generalization of the van der Waerden numbers. Ap _k -sequence of length n is defined to be a strictly increasing sequence of positive integers {x ₁, ...,x _n } for which there exists a polynomial of degree at mostk with integer coefficients and satisfyingf(x _j )=x _j+1 forj=1, 2, ...,n?1. Definep _k (n) to be the least positive integer such that if {1, 2, ...,p _k (n)} is partitioned into two sets, then one of the two sets must contain ap _k -sequence of lengthn. THEOREM. p_n?2(n)≦(n!)^(n?2)!/2.     

On cubic non‐Cayley vertex‐transitive graphs

In 1983, the second author [D. Maru?i?, Ars Combinatoria 16B (1983), 297–302] asked for which positive integers n there exists a non‐Cayley vertex‐transitive graph on n vertices. (The term non‐Cayley numbers has later been given to such integers.) Motivated by this problem, Feng [Discrete Math 248 (2002), 265–269] asked to determine the smallest valency ?(n) among valencies of non‐Cayley vertex‐transitive graphs of order n. As cycles are clearly Cayley graphs, ?(n)?3 for any non‐Cayley number n. In this paper a goal is set to determine those non‐Cayley numbers n for which ?(n) = 3, and among the latter to determine those for which the generalized Petersen graphs are the only non‐Cayley vertex‐transitive graphs of order n. It is known that for a prime p every vertex‐transitive graph of order p, p² or p³ is a Cayley graph, and that, with the exception of the Coxeter graph, every cubic non‐Cayley vertex‐transitive graph of order 2p, 4p or 2p² is a generalized Petersen graph. In this paper the next natural step is taken by proving that every cubic non‐Cayley vertex‐transitive graph of order 4p², p>7 a prime, is a generalized Petersen graph. In addition, cubic non‐Cayley vertex‐transitive graphs of order 2p^k, where p>7 is a prime and k?p, are characterized. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 77–95, 2012     

Limit Laws for Sums of Independent Random Products: the Lattice Case

Let {V _i,j ;(i,j)∈ℕ²} be a two-dimensional array of independent and identically distributed random variables. The limit laws of the sum of independent random products

A degree condition for the circumference of a graph

We present a new condition on the degree sums of a graph that implies the existence of a long cycle. Let c(G) denote the length of a longest cycle in the graph G and let m be any positive integer. Suppose G is a 2-connected graph with vertices x₁,…,x_n and edge set E that satisfies the property that, for any two integers j and k with j < k, x_jx_k ? E, d(x_i) ? j and d(x_k) ? K - 1, we have (1) d(x_i) + d(x_k ? m if j + k ? n and (2) if j + k < n, either m ? n or d(x_j) + d(x_k) ? min(K + 1,m). Then c(G) ? min(m, n). This result unifies previous results of J.C. Bermond and M. Las Vergnas, respectively.     

Distance-balanced graphs: Symmetry conditions

A graph X is said to be distance-balanced if for any edge uv of X, the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. A graph X is said to be strongly distance-balanced if for any edge uv of X and any integer k, the number of vertices at distance k from u and at distance k+1 from v is equal to the number of vertices at distance k+1 from u and at distance k from v. Exploring the connection between symmetry properties of graphs and the metric property of being (strongly) distance-balanced is the main theme of this article. That a vertex-transitive graph is necessarily strongly distance-balanced and thus also distance-balanced is an easy observation. With only a slight relaxation of the transitivity condition, the situation changes drastically: there are infinite families of semisymmetric graphs (that is, graphs which are edge-transitive, but not vertex-transitive) which are distance-balanced, but there are also infinite families of semisymmetric graphs which are not distance-balanced. Results on the distance-balanced property in product graphs prove helpful in obtaining these constructions. Finally, a complete classification of strongly distance-balanced graphs is given for the following infinite families of generalized Petersen graphs: GP(n,2), GP(5k+1,k), GP(3k±3,k), and GP(2k+2,k).     

A generalization of the Gagliardo inequality

Consider functions u₁, u₂,..., u_n ∈ D(ℝ^k) and assume that we are given a certain set of linear combinations of the form ∑_{i, j} a _ij ^(l) ∂_ju_i. Sufficient conditions in terms of coefficients a _ij ^(l) are indicated under which the norms are controlled in terms of the L¹-norms of these linear combinations. These conditions are mostly transparent if k = 2. The classical Gagliardo inequality corresponds to a single function u₁ = u and the collection of its partial derivatives ∂₁u,..., ∂_ku. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 345, 2007, pp. 120–139.     

Resolvability in circulant graphs

A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v ∈ V (G) there is a vertex w ∈ W such that d(u, w) ≠ d(v, w). A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim(G). For a vertex u of G and a subset S of V (G), the distance between u and S is the number min s∈S d(u, s). A k-partition Π = {S 1 , S 2 , . . . , S k } of V (G) is called a resolving partition if for every two distinct vertices u, v ∈ V (G) there is a set S i in Π such that d(u, Si )≠ d(v, Si ). The minimum k for which there is a resolving k-partition of V (G) is called the partition dimension of G, denoted by pd(G). The circulant graph is a graph with vertex set Zn , an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i-j (mod n) ∈ C , where CZn has the property that C =-C and 0 ■ C. The circulant graph is denoted by Xn, Δ where Δ = |C|. In this paper, we study the metric dimension of a family of circulant graphs Xn, 3 with connection set C = {1, n/2 , n-1} and prove that dim(Xn, 3 ) is independent of choice of n by showing that dim(Xn, 3 ) ={3 for all n ≡ 0 (mod 4), 4 for all n ≡ 2 (mod 4). We also study the partition dimension of a family of circulant graphs Xn,4 with connection set C = {±1, ±2} and prove that pd(Xn, 4 ) is independent of choice of n and show that pd(X5,4 ) = 5 and pd(Xn,4 ) ={3 for all odd n ≥ 9, 4 for all even n ≥ 6 and n = 7.