# EXTREMAL PROBLEMS CONCERNING CYCLES IN GRAPHS AND THEIR COMPLEMENTS

Kusmayadi, Tri Atmojo and Caccetta, L (2013) EXTREMAL PROBLEMS CONCERNING CYCLES IN GRAPHS AND THEIR COMPLEMENTS. Universitas Sebelas Maret. PDF (EXTREMAL PROBLEMS CONCERNING CYCLES IN GRAPHS AND THEIR COMPLEMENTS ) Download (178Kb)

## Abstract

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, circumference

Item Type: Other Q Science > QA Mathematics Fakultas Matematika dan Ilmu Pengetahuan Alam > Matematika mr azis r 22 May 2013 20:13 17 Mar 2017 12:06 https://eprints.uns.ac.id/id/eprint/829