Minimal size of a graph with diameter 2 and given maximal degree. II.
The paper is devoted to estimates of the spanning tree congestion for grid graphs and discrete toruses of dimensions two and three.
We describe two constructions of (very) dense graphs which are edge disjoint unions of large induced matchings. The first construction exhibits graphs on vertices with edges, which can be decomposed into pairwise disjoint induced matchings, each of size . The second construction provides a covering of all edges of the complete graph by two graphs, each being the edge disjoint union of at most induced matchings, where . This disproves (in a strong form) a conjecture of Meshulam, substantially...
Impartial Solitaire Clobber is a one-player version of the combinatorial game Clobber, introduced by Albert et al. in 2002. The initial configuration of Impartial Solitaire Clobber is a graph, such that there is a stone placed on each of its vertex, each stone being black or white. A move of the game consists in picking a stone, and clobbering an adjacent stone of the opposite color. By clobbering we mean that the clobbered stone is removed from the graph, and replaced by the clobbering one....