William Pettersson

Cycle decompositions V: Complete graphs into cycles of arbitrary lengths

We show that the complete graph on n vertices can be decomposed into t cycles of specified lengths m1,..., mt if and only if n is odd, 3≤mi≤n for i=1,..., t, and. We also show that the complete graph on n vertices can be decomposed into a perfect matching and t cycles of specified lengths m1,..., mt if and only if n is even, 3≤mi≤n for i=1,..., t, and m1 + ··· + mt = n2- n/2.

Bipartite 2-Factorizations of Complete Multipartite Graphs

It is shown that if K is any regular complete multipartite graph of even degree, and F is any bipartite 2-factor of K, then there exists a factorization of K into F; except that there is no factorization of K6, 6 into F when F is the union of two disjoint 6-cycles.

Bipartite 2-ractorizations of complete multipartite graphs

It is shown that if K is any regular complete multipartite graph of even degree, and F is any bipartite 2-factor of K, then there exists a factorization of K into F; except that there is no factorization of K6, 6 into F when F is the union of two disjoint 6-cycles.

A parallel approach to bi-objective integer programming

The real world applications of optimisation algorithms often are only interested in the running time of an algorithm, which can frequently be significantly reduced through parallelisation. We present two methods of parallelising the recursive algorithm presented by Ozlen, Burton and MacRae [J. Optimization Theory and Applications; 160:470--482, 2014]. Both new methods utilise two threads and improve running times. One of the new methods, the Meeting algorithm, halves running time to achieve near-perfect parallelisation, allowing users to solve bi-objective integer problems with more variables.