Mateja Šajna

Cycle decompositions IV: complete directed graphs and fixed length directed cycles

We establish necessary and sufficient conditions for decomposing the complete symmetric digraph of order n into directed cycles of length m, where 2≤m≤n.

Half-transitivity of some metacirculants

Abstract In 1991, Alspach, Marusic, and Nowitz proved that there are infinitely many 12-transitive graphs of degree 4. Their graphs were found among metacirculants M(α; m, n), which have vertex set {vji: i ϵ Zm, j ϵ Zn} and edge set {vji vj+δαii+1: i ϵ Zn, δ ϵ {− 1, 1}} with the additional condition that α ∈ Z n ∗ has order m or 2m. Examining only the cases when both m and n are odd, they showed that the graphs M(α; 3, n) are 12-transitive when n ⩾ 9 and gave a sufficient condition for M(α; m, n) to be 12-transitive when m is composite and n is prime. In this paper, we give a simple generalization of this condition. We also show that the graphs M(α; 2, n) are arc-transitive. Then we examine the graphs M(α; 4, n). We prove that they are arc-transitive when the order of α is 4 with α2 ≡ −1 (mod n) and 12-transitive when either the order of α is 8 or the order of α is 4 with α2 ≢ −1 (mod n) and n is not a multiple of 4.

Almost self-complementary circulant graphs

An almost self-complementary graph is a graph isomorphic to its complement minus a 1-factor. An almost self-complementary circulant graph is called cyclically almost self-complementary if it has an isomorphic almost complement with the same regular cyclic subgroup of the automorphism group. In this paper we prove that a cyclically almost self-complementary circulant of order 2n exists if and only if every prime divisor of n is congruent to 1 modulo 4, thus extending the known result on the existence of self-complementary circulants. We also describe the structure of cyclically almost self-complementary circulants and the action of their automorphism groups. Finally, we exhibit a class of almost self-complementary Cayley graphs on a dihedral group that are isomorphic to cyclically almost self-complementary circulants.

On decomposing K n -I into cycles of a fixed odd length

We show that if Kn - I, the complete graph on n vertices with a 1-factor removed, can be decomposed into cycles of a fixed odd length m for all n in the range m ≤ n > 3m that satisfy the obvious necessary conditions, then Kn - I can be decomposed into m-cycles for all n ≥ m satisfying the necessary conditions.

Self-complementary two-graphs and almost self-complementary double covers

A graph X is called almost self-complementary with respect to a perfect matching I if it is isomorphic to the graph obtained from its complement X^c by removing the edges of I. A two-graph on a vertex set @W is a collection T of 3-subsets of @W such that each 4-subset of @W contains an even number of elements of T. In this paper we investigate the relationship between self-complementary two-graphs and double covers over complete graphs that are almost self-complementary with respect to a set of fibres. In particular, we classify all doubly transitive self-complementary two-graphs, and thus all almost self-complementary graphs with an automorphism group acting 2-transitively on the corresponding perfect matching.

Vertex-transitive self-complementary uniform hypergraphs

In this paper we examine the orders of vertex-transitive self-complementary uniform hypergraphs. In particular, we prove that if there exists a vertex-transitive self-complementary k-uniform hypergraph of order n, where k=2^@? or k=2^@?+1 and n=1(mod2^@?^+^1), then the highest power of any prime dividing n must be congruent to 1 modulo 2^@?^+^1. We show that this necessary condition is also sufficient in many cases-for example, for n a prime power, and for k=3 and n odd-thus generalizing the result on vertex-transitive self-complementary graphs of Rao and Muzychuk. We also give sufficient conditions for the existence of vertex-transitive self-complementary uniform hypergraphs in several other cases. Since vertex-transitive self-complementary uniform hypergraphs are equivalent to a certain kind of large sets of t-designs, the results of the paper imply the corresponding results in design theory.

On almost self-complementary graphs

A graph is called almost self-complementary if it is isomorphic to one of its almost complements X^c-I, where X^c denotes the complement of X and I a perfect matching (1-factor) in X^c. Almost self-complementary circulant graphs were first studied by Dobson and Sajna [Almost self-complementary circulant graphs, Discrete Math. 278 (2004) 23-44]. In this paper we investigate some of the properties and constructions of general almost self-complementary graphs. In particular, we give necessary and sufficient conditions on the order of an almost self-complementary regular graph, and construct infinite families of almost self-complementary regular graphs, almost self-complementary vertex-transitive graphs, and non-cyclically almost self-complementary circulant graphs.

Homogeneously almost self-complementary graphs

Abstract We give necessary and sufficient conditions on n for there to exist a homogeneously almost self-complementary graph of order 2 n for the cases when n is an odd prime power, and when n is twice a prime. We introduce a unified approach to almost self-complementary graphs via brick assignments, and use it to construct several important families of homogeneously almost self-complementary graphs.

The women made it work: fuzzy transitive closure of the results chain in a dengue prevention trial in Mexico

BackgroundA modified theory of planned behaviour (acronym CASCADA) proposes that Conscious knowledge precedes a change in Attitude, which in turn precedes positive deviations from negative Subjective norms, intention to Change, perception of Agency to change, Discussion of possible action, and Action itself. We used this as a results chain to investigate gender-specific behaviour dynamics in chemical-free dengue prevention.MethodsSecondary analysis of the Mexican arm of a cluster randomised controlled trial used household survey data on intermediate outcomes of dengue prevention behaviour. We used a matrix of odds ratios between outcomes, transformed to a symmetrical range (−1, 1), to compute fuzzy transitive closure of the results chain for control and intervention clusters, then for male and female respondents separately in each group. Transitive closure of a map computes the influence of each factor on each other factor, taking account of all influences in the system. Cumulative net influence was the sum of influences across the results chain.ResultsResponses of 5042 women and 1143 men in 45 intervention clusters contrasted with those of 5025 women and 1179 men in 45 control clusters. Control clusters showed a distal block (negative influence) in the results chain with a cumulative net influence of 0.88; intervention clusters showed no such block and a cumulative net influence of 1.92. Female control respondents, like the overall control picture, showed a distal block, whereas female intervention responses showed no such blocks (cumulative net influence 0.78 and 1.73 respectively). Male control respondents showed weak distal blocks. Male intervention responses showed several new negative influences and a reduction of cumulative net influence (1.38 in control and 1.11 in intervention clusters).ConclusionsThe overall influence of the intervention across the results chain fits with the trial findings, but is different for women and men. Among women, the intervention overcame blocks and increased the cumulative net influence of knowledge on action. Among men, the intervention did not reinforce prevention behaviour. This might be related to emphasis, during the intervention, on women’s participation and empowerment. The fuzzy transitive closure of the CASCADA map usefully highlights the differences between gender-specific results chains.Trial registrationISRCTN27581154.

Computing transitive closure of bipolar weighted digraphs

We define a bipolar weighted digraph as a weighted digraph together with the sign function on the arcs such that the weight of each arc lies between 0 and 1, and no two parallel arcs have the same sign. Bipolar weighted digraphs are utilized to model so-called fuzzy cognitive maps, which are used in science, engineering, and the social sciences to represent the causal structure of a body of knowledge. It has been noted in the literature that a transitive closure of a bipolar weighted digraph contains useful new information for the fuzzy cognitive map it models. In this paper we ask two questions: what is a sensible and useful definition of transitive closure of a bipolar weighted digraph, and how do we compute it? We give two answers to each of these questions, that is, we present two distinct models. First, we give a review of the fuzzy digraph model, which has been, in a different form and less rigorously, studied previously in the fuzzy systems literature. Second, we carefully develop a probabilistic model, which is related to the notion of network reliability. This paper is intended for a mathematical audience.