EXTREMAL PROBLEMS CONCERNING CYCLES IN GRAPHS AND THEIR COMPLEMENTS

Let Gt(n) be the class of connected graphs on n vertices having the longest cycle of length t and let G ∈ Gt(n). Woodall (1976) determined the maximum number of edges of G, ε(G) ≤ w(n,t), where w(n, t) = (n - 1) t/2 - r(t – r - 1)/2 and r = (n - 1 ) - (t - 1) ⎣(n - 1)/(t - 1)⎦. An alternative proof and characterization of the extremal (edge-maximal) graphs given by Caccetta and Vijayan (1991). The edge- maximal graphs have the property that their complements are either disconnected or have a cycle going through each vertex (i.e. they are hamiltonian). This motivates us to investigate connected graphs with prescribed circumference (length of the longest cycle) having connected complements with cycles . More specifically, we focus our investigations on : Let G(n, c, c ) denote the class of connected graphs on n vertices having circumference c and whose connected complements have circumference c . The problem of interest is that of determining the bounds of the number of edges of a graph G ∈ G(n, c, c ) and characterize the extremal graphs of G(n, c, c ). We discuss the class G(n, c, c ) and present some results for small c. In particular for c = 4 and c = n - 2, we provide a complete solution. Key words : extremal graph, circumferenc

Minimum Cycle Base of Graphs Identified by Two Planar Graphs

In this paper, we study the minimum cycle base of the planar graphs obtained from two 2-connected planar graphs by identifying an edge (or a cycle) of one graph with the corresponding edge (or cycle) of another, related with map geometries, i.e., Smarandache 2-dimensional manifolds

Graphs that do not contain a cycle with a node that has at least two neighbors on it

We recall several known results about minimally 2-connected graphs, and show that they all follow from a decomposition theorem. Starting from an analogy with critically 2-connected graphs, we give structural characterizations of the classes of graphs that do not contain as a subgraph and as an induced subgraph, a cycle with a node that has at least two neighbors on the cycle. From these characterizations we get polynomial time recognition algorithms for these classes, as well as polynomial time algorithms for vertex-coloring and edge-coloring