Tactical decompositions and some configurationsv 4

The study of configurations or — more generally — finite incidence geometries is best accomplished by taking into account also their automorphism groups. These groups act on the geometry and in particular on points, blocks, flags and even anti-flags. The orbits of these groups lead to tactical decompositions of the incidence matrices of the geometries or of related geometries. We describe the general procedure and use these decompositions to study symmetric configurationsv ₄ for smallv. Tactical decompositions have also been used to construct all linear spaces on 12 points [2] and all proper linear spaces on 17 points [3].     

ON CIRCULANT DIGRAPHS WITH REGULAR AUTOMORPHISM GROUPS

Abstract. We prove that the cyclic group Zn(n≥3) has a k-regular digraph regular     

The partition dimension of circulant graphs

Let Π = {S₁, S₂, . . . , S_k} be an ordered partition of the vertex set V (G) of a graph G. The partition representation of a vertex v ∈ V (G) with respect to Π is the k-tuple r(v|Π) = (d(v, S₁), d(v, S₂), . . . , d(v, S_k)), where d(v, S) is the distance between v and a set S. If for every pair of distinct vertices u, v ∈ V (G), we have r(u|Π) ≠ r(v|Π), then Π is a resolving partition and the minimum cardinality of a resolving partition of V (G) is called the partition dimension of G. We study the partition dimension of circulant graphs, which are Cayley graphs of cyclic groups. Grigorious et al. [On the partition dimension of circulant graphs] proved that pd(C_n(1, 2, . . . , t)) ≥ t + 1 for n ≥ 3. We disprove this statement by showing that if t ≥ 4 is even, then there exists an infinite set of values of n, such that . We also present exact values of the partition dimension of circulant graphs with 3 generators.     

Graph Connectivity After Path Removal

Let G be a graph and u, v be two distinct vertices of G. A u—v path P is called nonseparating if G—V(P) is connected. The purpose of this paper is to study the number of nonseparating u—v path for two arbitrary vertices u and v of a given graph. For a positive integer k, we will show that there is a minimum integer (k) so that if G is an (k)-connected graph and u and v are two arbitrary vertices in G, then there exist k vertex disjoint paths P ₁[u,v], P ₂[u,v], . . ., P _k [u,v], such that G—V (P _i [u,v]) is connected for every i (i = 1, 2, ..., k). In fact, we will prove that (k) 22k+2. It is known that (1) = 3.. A result of Tutte showed that (2) = 3. We show that (3) = 6. In addition, we prove that if G is a 5-connected graph, then for every pair of vertices u and v there exists a path P[u, v] such that G—V(P[u, v]) is 2-connected.* Supported by NSF grant No. DMS-0070059 Supported by ONR grant N00014-97-1-0499 Supported by NSF grant No. 9531824     

The polytope of win vectors

Imagine a graph as representing a fixture list with vertices corresponding to teams, and the number of edges joiningu andv as representing the number of games in whichu andv have to play each other. Each game ends in a win, loss, or tie and we say a vector =(w ₁,...,w _n) is awin vector if it represents the possible outcomes of the games, withw _i denoting the total number of games won by teami. We study combinatorial and geometric properties of the set of win vectors and, in particular, we consider the problem of counting them. We construct a fully polynomial randomized approximation scheme for their number in dense graphs. To do this we prove that the convex hull of the set of win vectors ofG forms an integral polymatroid and then use volume approximation techniques. Supported by the “DAAD Doktorandenstipendium des zweiten Hochschulsonderprogrammes HSPII/AUFE”. Partially supported by RAND-REC EC US030.     

W v cycles in plane graphs

A cycle in a plane graphG is called aW _v cycle if it has a connected (or empty) intersection with each face of the graph. We show that if the minimum degree (G)3 thenG has aW _v cycle and the lengthw(G) of a longestW _v cycle is bounded by the number,f(G), of faces ofG. The classW of graphsG withw(G)=f(G) is completely characterized by an characterized by an inductive construction from two graphs, namelyK ₄ and a face merging of two copies ofK ₄ on one hand, and in terms involving Halin graphs and face merging on the other hand. Longest cycles in members ofW are investigated. The shortness coefficient ofW is proved to be between one-half and three-quarters inclusively.     

ON A CONJECTURE ON nTH ORDER DEGREE REGULAR GRAPHS

Abstract

For n a positive integer and v a vertex of a graph G, the nth order degree of v in G, denoted by deg_nv, is the number of vertices at distance n from v. The graph G is said to be nth order regular of degree k if, for every vertex v of G, deg_nv = k. The following conjecture due to Alavi, Lick, and Zou is proved: For n ≥ 2, if G is a connected nth order regular graph of degree 1, then G is either a path of length 2n—1 or G has diameter n. Properties of nth order regular graphs of degree k, k ≥ 1, are investigated.     

The 2‐Blocking Number and the Upper Chromatic Number of PG(2,q)

A twofold blocking set (double blocking set) in a finite projective plane Π is a set of points, intersecting every line in at least two points. The minimum number of points in a double blocking set of Π is denoted by τ₂(Π). Let PG(2,q) be the Desarguesian projective plane over GF(q), the finite field of q elements. We show that if q is odd, not a prime, and r is the order of the largest proper subfield of GF(q), then τ₂PG(2,q))≤ 2(q+(q‐1)/(r‐1)). For a finite projective plane Π, let denote the maximum number of classes in a partition of the point‐set, such that each line has at least two points in some partition class. It can easily be seen that (?) for every plane Π on v points. Let , p prime. We prove that for , equality holds in (?) if q and p are large enough.     

On the number of lines in planar spaces

SupposeS is a planar space withv>4 points and letq be the positive real number such thatv=q ³+q²+q+1. Assuming a weak non-degeneracy condition, we shall show thatS has at least (q²+1)(q²+q+1) lines with equality iffq is a prime power andS=PG(3,q).     

Non-Axiomatizability of the amalgamation class of modular lattice varieties

We prove that if v is the variety generated by a finite modular lattice, then v is not an elementary class. We also consider the same question for the variety generated by N ₅.Research partially supported by National Science Foundation grant DMS-8701643.     

Resolvable Mendelsohn designs with block size 4

Summary Letv andK be positive integers. A (v, k, 1)-Mendelsohn design (briefly (v, k, 1)-MD) is a pair (X,B) whereX is av-set (ofpoints) andB is a collection of cyclically orderedk-subsets ofX (calledblocks) such that every ordered pair of points ofX are consecutive in exactly one block ofB. A necessary condition for the existence of a (v, k, 1)-MD isv(v–1) 0 (modk). If the blocks of a (v, k, 1)-MD can be partitioned into parallel classes each containingv/k blocks wherev ) (modk) or (v – 1)/k blocks wherev 1 (modk), then the design is calledresolvable and denoted briefly by (v, k, 1)-RMD. It is known that a (v, 3,1)-RMD exists if and only ifv 0 or 1 (mod 3) andv 6. In this paper, it is shown that the necessary condition for the existence of a (v, 4, 1)-RMD, namelyv 0 or 1 (mod 4), is also sufficient, except forv = 4 and possibly exceptingv = 12. These constructions are equivalent to a resolvable decomposition of the complete symmetric directed graphK _v ^* onv vertices into 4-circuits.Research supported by the Natural Sciences and Engineering Research Council of Canada under Grant A-5320.     

Note on a conjecture of toft

A conjecture of Toft [17] asserts that any 4-critical graph (or equivalently, every 4-chromatic graph) contains a fully odd subdivision ofK ₄. We show that if a graphG has a degree three nodev such thatG-v is 3-colourable, then eitherG is 3-colourable or it contains a fully oddK ₄. This resolves Toft's conjecture in the special case where a 4-critical graph has a degree three node, which is in turn used to prove the conjecture for line-graphs. The proof is constructive and yields a polynomial algorithm which given a 3-degenerate graph either finds a 3-colouring or exhibits a subgraph that is a fully odd subdivision ofK ₄. (A graph is 3-degenerate if every subgraph has some node of degree at most three.)     

Roman and Total Domination

Abstract

A set S of vertices is a total dominating set of a graph G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set is the total domination number ^γt(G). A Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f (u)=0 is adjacent to at least one vertex v of G for which f (v)=2. The minimum of f (V (G))=∑_{u ∈ V (G)} f (u) over all such functions is called the Roman domination number γR (G). We show that γt(G) ≤ γR (G) with equality if and only if ^γt(G)=2γ(G), where γ(G) is the domination number of G. Moreover, we characterize the extremal graphs for some graph families.     

Global italian domination in graphs

Abstract

An Italian dominating function (IDF) on a graph G = (V, E) is a function f: V → {0, 1, 2} satisfying the condition that for every vertex v ∈ V (G) with f (v) = 0, either v is adjacent to a vertex assigned 2 under f, or v is adjacent to at least two vertices assigned 1. The weight of an IDF f is the value ∑_v∈V(G) f (v). The Italian domination number of a graph G, denoted by γ_I (G), is the minimum weight of an IDF on G. An IDF f on G is called a global Italian dominating function (GIDF) on G if f is also an IDF on the complement ? of G. The global Italian domination number of G, denoted by γ_gI (G), is the minimum weight of a GIDF on G. In this paper, we initiate the study of the global Italian domination number and we present some strict bounds for the global Italian domination number. In particular, we prove that for any tree T of order n ≥ 4, γ_gI (T) ≤ γ_I (T) + 2 and we characterize all trees with γ_gI (T) = γ_I (T) + 2 and γ_gI (T) = γ_I (T) + 1.     

UPPER BOUNDS ON THE LOWER OPEN PACKING NUMBER OF A TREE

Abstract

Let G be a graph and let v be a vertex of G. The open neigbourhood N(v) of v is the set of all vertices adjacent with v in G. An open packing of G is a set of vertices whose open neighbourhoods are pairwise disjoint. The lower open packing number of G, denoted ρ^° _L(G), is the minimum cardinality of a maximal open packing of G while the (upper) open packing number of G, denoted ρ^°(G), is the maximum cardinality among all open packings of G. It is known (see [7]) that if G is a connected graph of order n ≥3, then ρ^°(G) ≤ 2n/3 and this bound is sharp (even for trees). As a consequence of this result, we know that ρ^° _L(G) ≤ 2n/3. In this paper, we improve this bound when G is a tree. We show that if G is a tree of order n with radius 3, then ρ^° _L(G) ≤ n/2 + 2 √n-1, and this bound is sharp, while if G is a tree of order n with radius at least 4, then ρ^° _L(G) is bounded above by 2n/3—O√n).     

Free lattice algorithms

In the late 1930s Phillip Whitman gave an algorithm for deciding for lattice terms v and u if vu in the free lattice on the variables in v and u. He also showed that each element of the free lattice has a shortest term representing it and this term is unique up to commutivity and associativity. He gave an algorithm for finding this term. Almost all the work on free lattices uses these algorithms. Building on the work of Ralph McKenzie, J. B. Nation and the author have developed very efficient algorithms for deciding if a lattice term v has a lower cover (i.e., if there is a w with w covered by v, which is denoted by w) and for finding them if it does. This paper studies the efficiency of both Whitman's algorithm and the algorithms of Freese and Nation. It is shown that although it is often quite fast, the straightforward implementation of Whitman's algorithm for testing vu is exponential in time in the worst case. A modification of Whitman's algorithm is given which is polynomial and has constant minimum time. The algorithms of Freese and Nation are then shown to be polynomial.     

Scheduling tasks with communication delays on parallel processors

LetP={v ₁,...,v _n } be a set ofn jobs to be executed on a set ofm identical machines. In many instances of scheduling problems, if a jobv _i has to be executed before the jobv _j and both jobs are to be executed on different machines, some sort of information exchange has to take place between the machines executing them. The time it takes for this exchange of information is called a communication delay.In this paper we give anO(n) algorithm to find an optimal scheduling with communication delays when the number of machines is not limited and the precedence constraints on the jobs form a tree.     

Resolvable and near-resolvable decompositions ofDK v into oriented 4-cycles

Summary A resolvableX-decomposition ofDK _v (the complete symmetric digraph onv vertices) is a partition of the arcs ofDK _v into isomorphic factors where each factor is a vertex-disjoint union of copies ofX and spans all vertices ofDK _v . There are four orientations ofC ₄ (the 4-cycle), only one of which has been considered: Bennett and Zhang, Aequationes Math.40 (1990), 248–260. We give necessary and sufficient conditions onv for resolvableX-decomposition ofDK _v , whereX is any one of the other three orientations ofC ₄. A near-resolvableX-decomposition ofDK _v is as above except that each factor spans all but one vertex ofDK _v . Again, one orientation ofC ₄ has been dealt with by Bennett and Zhang, and we provide necessary and sufficient conditions onv for the remaining three cases. The construction techniques used are both direct (for small values ofv) and recursive.The author thanks Simon Fraser University for its support during her graduate studies when the research for this paper was undertaken.The author acknowledges the Natural Sciences and Engineering Research Council of Canada for financial support under grant A-7829.     

Almost Covers Of 2-Arc Transitive Graphs

Let be a G-symmetric graph whose vertex set admits a nontrivial G-invariant partition with block size v. Let be the quotient graph of relative to and [B,C] the bipartite subgraph of induced by adjacent blocks B,C of . In this paper we study such graphs for which is connected, (G, 2)-arc transitive and is almost covered by in the sense that [B,C] is a matching of v-1 2 edges. Such graphs arose as a natural extremal case in a previous study by the author with Li and Praeger. The case K _v+1 is covered by results of Gardiner and Praeger. We consider here the general case where K _v+1, and prove that, for some even integer n 4, is a near n-gonal graph with respect to a certain G-orbit on n-cycles of . Moreover, we prove that every (G, 2)-arc transitive near n-gonal graph with respect to a G-orbit on n-cycles arises as a quotient of a graph with these properties. (A near n-gonal graph is a connected graph of girth at least 4 together with a set of n-cycles of such that each 2-arc of is contained in a unique member of .)     

Tournaments with prescribed regular automorphism group

A tournamentX is a TRR for a groupG if (a)G acts regularly on the vertices ofX and (b) Aut(X) is isomorphic toG. We correct some previous work of Babai and Imrich by showing thatZ ₂ ³ andZ ₃ ³ are the only groups of odd order without TRR's. Our methods are perhaps of independent interest, since we use a probabilistic approach.