33185867
The computational complexity of the parallel knock-out problem
We consider computational complexity questions related to parallel knock-out schemes for graphs. In such schemes, in each round, each remaining vertex of a given graph eliminates exactly one of its neighbours. We show that the problem of whether, for a given bipartite graph, such a scheme can be found that eliminates every vertex is NP-complete. Moreover, we show that, for all fixed positive integers k>=2, the problem of whether a given bipartite graph admits a scheme in which all vertices are eliminated in at most (exactly) k rounds is NP-complete. For graphs with bounded tree-width, however, both of these problems are shown to be solvable in polynomial time. We also show that r-regular graphs with r>=1, factor-critical graphs and 1-tough graphs admit a scheme in which all vertices are eliminated in one round.
40777411
Amalgamations of connected k-factorizations
In this paper, necessary and sufficient conditions are found for a graph with exactly one amalgamated vertex to be the amalgamation of a k-factorization of Kkn+1 in which each k factor is connected. From this result, necessary and sufficient conditions for a given edge-coloured Kt to be embedded in a connected k-factorization of Kkn+1 are deduced.
7413347
The Computational Complexity of the Parallel Knock-Out Problem
We consider computational complexity questions related to parallel knock-out schemes for graphs. In such schemes, in each round, each remaining vertex of a given graph eliminates exactly one of its neighbours. We show that the problem of whether, for a given graph, such a scheme can be found that eliminates every vertex is NP-complete. Moreover, we show that, for all fixed positive integers k ≥ 2, the problem of whether a given graph admits a scheme in which all vertices are eliminated in at most k rounds is NP-complete. For graphs with bounded tree-width, however, both of these problems are shown to be solvable in polynomial time.
41320428
Characterization of graphs with hall number 2
Hall’s condition is a simple condition that a graph G and list assignment L must satisfy if G has a proper L-colouring. The Hall number of G is the smallest integer m such that whenever each list has size m and Hall’s condition is satisfied a proper L-colouring exists. Hilton and P.D. Johnson introduced the parameter and showed that a graph has Hall number 1 if and only if every block is a clique. In this paper we characterize graphs with Hall number 2. ∗Current address: London School of Economics, Houghton Street, London WC2A 2AE, U.K. e-mail:matthew@maths.lse.ac.uk