Ten Exceptional Geometries from Trivalent Distance Regular Graphs

The ten distance regular graphs of valency 3 and girth > 4 define ten non-isomorphic neighborhood geometries, amongst which a projective plane, a generalized quadrangle, two generalized hexagons, the tilde geometry, the Desargues configuration and the Pappus configuration. All these geometries are bislim, i.e., they have three points on each line and three lines through each point. We study properties of these geometries such as embedding rank, generating rank, representation in real spaces, alternative constructions. Our main result is a general construction method for homogeneous embeddings of flag transitive self-polar bislim geometries in real projective space.     

Uniformly Packed Codes and More Distance Regular Graphs from Crooked Functions

Let V and W be n-dimensional vector spaces over GF(2). A function Q : V W is called crooked (a notion introduced by Bending and Fon-Der-Flaass) if it satisfies the following three properties:

Orthogonal Double Covers of Complete Graphs by Caterpillars of Diameter 5

An orthogonal double cover (ODC) of the complete graph K_n by a graph G is a collection = {G_i|i = 1,2, . . . ,n} of spanning subgraphs of K_n, all isomorphic to G, with the property that every edge of K_n belongs to exactly two members of and any two distinct members of share exactly one edge. A caterpillar of diameter five is a tree arising from a path with six vertices by attaching pendant vertices to some or each of its vertices of degree two. We show that for any caterpillar of diameter five there exists an ODC of the complete graph K_n.     

Distance Magic Labeling in Complete 4-partite Graphs

Let G be a complete k-partite simple undirected graph with parts of sizes \(p_1\le p_2\cdots \le p_k\). Let \(P_j=\sum _{i=1}^jp_i\) for \(j=1,\ldots ,k\). It is conjectured that G has distance magic labeling if and only if \(\sum _{i=1}^{P_j} (n-i+1)\ge j{{n+1}\atopwithdelims (){2}}/k\) for all \(j=1,\ldots ,k\). The conjecture is proved for \(k=4\), extending earlier results for \(k=2,3\).     

Covers of Point-Hyperplane Graphs

A cover of the non-incident point-hyperplane graph of projective dimension 3 for fields of characteristic 2 is constructed. For fields of even order larger than 2, this leads to an elementary construction of the non-split extension of SL₄( )by ⁶.     

Elementary Abelian Covers of Graphs

Let _G (X) be the set of all (equivalence classes of) regular covering projections of a given connected graph X along which a given group G Aut X of automorphisms lifts. There is a natural lattice structure on _G (X), where _{1 2} whenever ₂ factors through ₁. The sublattice _G () of coverings which are below a given covering : X^~ X naturally corresponds to a lattice _G () of certain subgroups of the group of covering transformations. In order to study this correspondence, some general theorems regarding morphisms and decomposition of regular covering projections are proved. All theorems are stated and proved combinatorially in terms of voltage assignments, in order to facilitate computation in concrete applications.For a given prime p, let _G ^p (X) _G (X) denote the sublattice of all regular covering projections with an elementary abelian p-group of covering transformations. There is an algorithm which explicitly constructs _G ^p (X) in the sense that, for each member of _G ^p (X), a concrete voltage assignment on X which determines this covering up to equivalence, is generated. The algorithm uses the well known algebraic tools for finding invariant subspaces of a given linear representation of a group. To illustrate the method two nontrival examples are included.     

Latin Squares, Partial Latin Squares and Their Generalized Quotients

A (partial) Latin square is a table of multiplication of a (partial) quasigroup. Multiplication of a (partial) quasigroup may be considered as a set of triples. We give a necessary and sufficient condition for a set of triples to be a quotient of a (partial) Latin square. Received: April, 2003     

A Class of 2-Colorable Orthogonal Double Covers of Complete Graphs by Hamiltonian Paths

K _n by a graph G is a collection ? of n spanning subgraphs of K _n , all isomorphic to G, such that any two members of ? share exactly one edge and every edge of K _n is contained in exactly two members of ?. In the 1980's Hering posed the problem to decide the existence of an ODC for the case that G is an almost-hamiltonian cycle, i.e. a cycle of length n−1. It is known that the existence of an ODC of K _n by a hamiltonian path implies the existence of ODCs of K _4n and K _16n , respectively, by almost-hamiltonian cycles. Horton and Nonay introduced 2-colorable ODCs and showed: If for n≥3 and a prime power q≥5 there are an ODC of K _n by a hamiltonian path and a 2-colorable ODC of K _q by a hamiltonian path, then there is an ODC of K _qn by a hamiltonian path. We construct 2-colorable ODCs of K _n and K _2n , respectively, by hamiltonian paths for all odd square numbers n≥9. Received: January 27, 2000     

正交拉丁方组和泛对角线幻方的一种构造法

利用线性取余变换构造素数阶完备正交拉丁方组,给出泛对角线幻方的一种构造法.     

On Orthogonal Double Covers of Graphs

A transformation which allows us to obtain an orthogonal double cover of a graph G from any permutation of the edge set of G is described. This transformation is used together with existence results for self-orthogonal latin squares, to give a simple proof of a conjecture of Chung and West.     

Shortest Circuit Covers of Cubic Graphs

We show that the edge set of a bridgeless cubic graph G can be covered with circuits such that the sum of the lengths of the circuits is at most |E(G)|. Stronger results are obtained for cubic graphs of large girth.     

On Non-Polynomial Latin Squares

It turns out that Latin squares which are hard to approximate by a polynomial are suitable to be used as a part of block cipher algorithms (BCA). In this paper we state basic properties of those Latin squares and provide their construction.     

On Orthogonal Double Covers of Graphs

An orthogonal double cover (ODC) is a collection of n spanning subgraphs(pages) of the complete graph K _n such that they cover every edge of the completegraph twice and the intersection of any two of them contains exactly one edge. If all the pages are isomorphic tosome graph G, we speak of an ODC by G. ODCs have been studied for almost 25 years, and existenceresults have been derived for many graph classes. We present an overview of the current state of research alongwith some new results and generalizations. As will be obvious, progress made in the last 10 years is in many waysrelated to the work of Ron Mullin. So it is natural and with pleasure that we dedicate this article to Ron, on theoccasion of his 65th birthday.     

Distance Integral Complete Multipartite Graphs with s=5, 6

Let D(G) = (dij )n×n denote the distance matrix of a connected graph G with order n, where dij is equal to the distance between vertices vi and vj in G. A graph is called distance integral if all eigenvalues of its distance matrix are integers. In 2014, Yang and Wang gave a su?cient and necessary condition for complete r-partite graphs Kp1,p2,··· ,pr =Ka1·p1,a2·p2,··· ,as···ps to be distance integral and obtained such distance integral graphs with s = 1, 2, 3, 4. However distance integral complete multipartite graphs Ka1·p1,a2·p2,··· ,as·ps with s>4 have not been found. In this paper, we find and construct some infinite classes of these distance integral graphs Ka1·p1,a2·p2,··· ,as·ps with s = 5, 6. The problem of the existence of such distance integral graphs Ka1·p1,a2·p2,··· ,as·ps with arbitrarily large number s remains open.     

Coloring Complete and Complete Bipartite Graphs from Random Lists

Assign to each vertex v of the complete graph \(K_n\) on n vertices a list L(v) of colors by choosing each list independently and uniformly at random from all f(n)-subsets of a color set \([n] = \{1,\dots , n\}\), where f(n) is some integer-valued function of n. Such a list assignment L is called a random (f(n), [n])-list assignment. In this paper, we determine the asymptotic probability (as \(n \rightarrow \infty \)) of the existence of a proper coloring \(\varphi \) of \(K_n\), such that \(\varphi (v) \in L(v)\) for every vertex v of \(K_n\). We show that this property exhibits a sharp threshold at \(f(n) = \log n\). Additionally, we consider the corresponding problem for the line graph of a complete bipartite graph \(K_{m,n}\) with parts of size m and n, respectively. We show that if \(m = o(\sqrt{n})\), \(f(n) \ge 2 \log n\), and L is a random (f(n), [n])-list assignment for the line graph of \(K_{m,n}\), then with probability tending to 1, as \(n \rightarrow \infty \), there is a proper coloring of the line graph of \(K_{m,n}\) with colors from the lists.     

Random Walks on Directed Covers of Graphs

Directed covers of finite graphs are also known as periodic trees or trees with finitely many cone types. We expand the existing theory of directed covers of finite graphs to those of infinite graphs. While the lower growth rate still equals the branching number, upper and lower growth rates no longer coincide in general. Furthermore, the behavior of random walks on directed covers of infinite graphs is more subtle. We provide a classification in terms of recurrence and transience and point out that the critical random walk may be recurrent or transient. Our proof is based on the observation that recurrence of the random walk is equivalent to the almost sure extinction of an appropriate branching process. Two examples in random environment are provided: homesick random walk on infinite percolation clusters and random walk in random environment on directed covers. Furthermore, we calculate, under reasonable assumptions, the rate of escape with respect to suitable length functions and prove the existence of the asymptotic entropy providing an explicit formula which is also a new result for directed covers of finite graphs. In particular, the asymptotic entropy of random walks on directed covers of finite graphs is positive if and only if the random walk is transient.     

Strong 5-Cycle Double Covers of Graphs

The Cycle Double Cover Conjecture claims that every bridgeless graph has a cycle double cover and the Strong Cycle Double Conjecture states that every such graph has a cycle double cover containing any specified circuit. In this paper, we get a necessary and sufficient condition for bridgeless graphs to have a strong 5-cycle double cover. Similar condition for the existence of 5-cycle double covers is also obtained. These conditions strengthen/improve some known results.     

Bipartite Graphs and Monochromatic Squares

Order - Let κ be a successor cardinal. We prove that consistently every bipartite graph of size κ+ × κ+ contains either an independent set or a clique of size τ ×...