In 1979, Nishizeki and Baybars showed that every planar graph with minimum degree 3 has a matching of size n + c (where the constant depends on connectivity), even better bounds hold for graphs 4 5. this paper, we investigate similar matching-bounds 1-planar graphs, is, can be drawn such edge at most one crossing. We show least 1 7 12 , is tight some graphs. provide 5, while case 6 remains open.

1. introduction climate is a large system which is produced due to the relation and action between other large systems. if a change occurs in one of these systems, other systems will coordinate themselves with the change fast or slowly. the consequences of this coordination will impress the system that is starter of this change and an endless continuum of the links connects these systems togeth...

An immersion of a graph H into a graph G is a one-to-one mapping f : V (H) → V (G) and a collection of edge-disjoint paths in G, one for each edge of H , such that the path Puv corresponding to edge uv has endpoints f(u) and f(v). The immersion is strong if the paths Puv are internally disjoint from f(V (H)). It is proved that for every positive integer t, every simple graph of minimum degree a...

The main result of this paper is as follows: Let G be a graph and m a positive integer such that (i) |V (G)| ≥ 4m+1, (ii) δ(G) ≥ |V (G)| 2 +1. For any subset {e1, . . . , em+1} of E(G), the graph G − {e1, . . . , em} possesses a 2-factor containing em+1. The above Theorem also yields a Dirac type sufficient condition for a graph to be magic. All graphs considered are assumed to be simple and fi...

We prove the following extension of an old result of Andrásfai, Erdős and Sós. For every fixed graph H with chromatic number r+1 ≥ 3, and for every fixed > 0, there are n0 = n0(H, ) and ρ = ρ(H) > 0, such that the following holds. Let G be an H-free graph on n > n0 vertices with minimum degree at least ( 1 − 1 r−1/3 + ) n. Then one can delete at most n2−ρ edges to make G r-colorable.

A graph is a triangulation if it is planar and every face is a triangle. A triangulation is rooted if the external triangular face is labelled. Two rooted triangulations with the same external face labels are isomorphic if their internal vertices can be labelled so that both triangulations have identical edge lists. In this article, we show that in the set of rooted triangulations on n points w...