A straight-line grid drawing of a planar graph G is a drawing of G on an integer grid such that each vertex is drawn as a grid point and each edge is drawn as a straight-line segment without edge crossings. It is well known that a planar graph of n vertices admits a straight-line grid drawing on a grid of area O(n). A lower bound of Ω(n) on the area-requirement for straight-line grid drawings o...

We propose three strategies for Presenter in on-line graph coloring games. The first one constructs bipartite graphs and forces any on-line coloring algorithm to use 2 log2 n − 10 colors, where n is the number of vertices in the constructed graph. This is best possible up to an additive constant. The second strategy constructs triangle-free graphs and forces Ω(n 1 2 ) colors. The third one cons...

We discover a surprising connection between graph coloring in two orthogonal paradigms: parallel and on-line computing. We present a randomized on-line coloring algorithm with a performance ratio of O(n= log n), an improvement of p log n factor over the previous best known algorithm of Vishwanathan. Also, from the same principles, we construct a parallel coloring algorithm with the same perform...

In one of fundamental work in combinatorial optimization Edmonds gave a complete linear description of the matching polytope. Matchings in a graph are equivalent to stable sets its line graph. Also the neighborhood of any vertex in a line graph partitions into two cliques: graphs with this latter property are called quasi-line graphs. Quasi-line graphs are a subclass of claw-free graphs, and as...

In this paper, we prove that a cubic line graph G on n vertices rather than the complete graph K4 has b3c vertex-disjoint triangles and the vertex independence number b3c. Moreover, the equitable chromatic number, acyclic chromatic number and bipartite density of G are 3, 3, 79 respectively.

For a graph G = (V; E) its line graph L(G) has the node set E and two nodes of L(G) are adjacent if the corresponding edges of G have a common endpoint. The problem of nding G for a given L was already optimally solved by Lehot7] and Roussopoulos10]. Here we present a new dynamic solution to this problem, where we can add or delete a node v in L(G) in time proportional to the size of its adjace...

For a given graph G, its line denoted by L(G) is whose vertex set V (L(G)) = E(G) and {e1, e2} ∈ E(L(G)) if e1 e2 are incident to common in G. Let R be finite commutative ring with nonzero identity G(R) denotes the unit associated R. In this manuscript, we have studied L(G(R)) of course investigation, several basic properties, viz., diameter, girth, clique, chromatic number been determined. Fur...