We first show that a graph is not bipartite if it contains any cycles of odd length. Starting at an arbitrary vertex u in this cycle, we note that there is a path of length 2k + 1 from u to v. We also note that to be bipartite, we must alternate placing vertices on this path into the partitions L and R. However, if we place u in L, then, we see that after following the path along 2k + 1 edges, ...

The interval number of a graph G is the least natural number t such that G is the intersection graph of sets, each of which is the union of at most t intervals, denoted by i (G ). Griggs and West showed that i(G) d12(d þ 1)e. We describe the extremal graphs for that inequality when d is even. For three special perfect graph classes we give bounds on the interval number in terms of the independe...

Let H,K be subgroups of G. We investigate the intersection properties of left and right cosets of these subgroups. If H and K are subgroups of G, then G can be partitioned as the disjoint union of all left cosets of H, as well as the disjoint union of all right cosets of K. But how do these two partitions of G intersect each other? Definition 1. Let G be a group, and H a subgroup of G. A left t...

Weighted finite-state machines with n tapes describe n-ary rational string relations. The join n-ary relation is very important regarding to applications. It is shown how to compute it via a more simple operation, the auto-intersection. Join and auto-intersection generally do not preserve rationality. We define a class of triples 〈A, i, j〉 such that the auto-intersection of the machine A w.r.t....

In this note, we give tighter bounds on the complexity of the bipartite unique perfect matching problem, bipartite-UPM. We show that the problem is in C=L and in NL , both subclasses of NC. We also consider the (unary) weighted version of the problem. We show that testing uniqueness of the minimum-weight perfect matching problem for bipartite graphs is in L= and in NL. Furthermore, we show that...

The interval number of a graph G, denoted i(G), is the least positive integer t for which G is the intersection graph of a family of sets each of which is the union of at most t cIosed intervals of the real line IR. Trotter and Harary showed that the interval number of the complete bipartite graph K(m, n) is [(mn + I)/(m + n)]. Matthews showed that the interval number of the complete multiparti...

We investigate the class of vertex intersection graphs of paths on a grid, and specifically consider the subclasses that are obtained when each path in the representation has at most k bends (turns). We call such a subclass the Bk-VPG graphs, k ≥ 0. If the number k of bends is not restricted, then the VPG graphs are shown to be equivalent to the well-known class of string graphs, namely, the in...

The minimum crossing number problem is among the oldest and most fundamental problems arising in the area of automatic graph drawing. In this paper, eight population-based meta-heuristic algorithms are utilized to tackle the minimum crossing number problem for two special types of graphs, namely complete graphs and complete bipartite graphs. A 2-page book drawing representation is employed for ...