We give upper and lower bounds on the size of mixed covering arrays on graphs based on graph homomorphisms. We provide constructions for covering arrays on graphs based on basic graph operations. In particular, we construct optimal mixed covering arrays on trees, cycles and bipartite graphs; the constructed optimal objects have the additional property of being nearly point balanced.

Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that a continuous quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time quantum walks on other well-behaved graphs do not exhibit this uniform mixing. We prove that the only graphs amongst balanced complete multipartite graphs th...

Our previous paper [9] applied a lopsided version of the Lovász Local Lemma that allows negative dependency graphs [5] to the space of random matchings in K2n, deriving new proofs to a number of results on the enumeration of regular graphs with excluded cycles through the configuration model [3]. Here we extend this from excluded cycles to some excluded balanced subgraphs, and derive asymptotic...

A signed graph (G,Σ) is a G together with set Σ⊆E(G) of negative edges. circuit positive if the product signs its edges positive. balanced all circuits are The frustration index l(G,Σ) minimum cardinality E⊆E(G) such that (G−E,Σ−E) balanced, and k-critical l(G,Σ)=k l(G−e,Σ−e)<k, for every e∈E(G). We study decomposition subdivision critical graphs completely determine t-critical graphs, t≤2. Cri...

Magda (1980) and Hedayat (1981) first considered the construction of circular strongly balanced repeated measurements designs. Sen and Mukerjee (1987) and Roy (1988) considered the optimality and existence of circular strongly balanced repeated measurements designs based on the method of differences and Hamiltonian decomposition of lexicographic product of two graphs. In this article, we consid...

By using the notion of compatibility of subgraphs with a perfect matching developed for digraphs in [1], we show that if, in a balanced bipartite graph G of minimum degree 6, the maximum cardinality ebip of a balanced independent subset satisfies ~bip ~< 26-4, then G is hamiltonian-biconnected, and if Ctbip ~< 26-2, G contains a hamiltonian path. Moreover, we give some properties of balanced bi...

For a hypergraph G with v vertices and e~ edges of size i, the average vertex degree is d(G) = =Eiedv. Call balanced if d(H)~-d(G) for all subhyoergraphs H of G. Let re(G0= max d(H). Hc=G A hypergraph F is said to be a balanced extension of G if GC F, F is balanced and tifF)-re(G), i.e. F is balanced and does not increase the maximum average degree. It is shown that for every hypergraph G there...

Galvin solved the Dinitz conjecture by proving that bipartite graphs are ∆edge-choosable. We improve Galvin’s method and deduce from any colouring of the edges of bipartite graph G some further list edge-colouring properties of G. In particular, for bipartite graphs, it follows from the existence of balanced bipartite edge-colourings that balanced list edge-colourings exist as well. While the k...

This paper classifies all finite edge colored graphs with doubly transitive automorphism groups. This result generalizes the classification of doubly transitive balanced incomplete block designs with 1 and doubly transitive one-factorizations of complete graphs. It also provides a classification of all doubly transitive symmetric association schemes. The classification of finite simple groups i...

The k-fold clique transversal problem is to locate a minimum set of vertices of a graph such that every maximal clique has at least k elements of. The maximum h-colourable subgraph problem is to nd a maximum subgraph of a graph which is h-colourable. We show that the k-fold clique transversal problem and the maximum h-colourable subgraph problem are polynomially solvable on balanced graphs. We ...