The quest for the eulerian recurrent lengths of complete bipartite graphs and complete graphs

Research output: Contribution to journalArticle

Abstract

The Eulerian recurrent length problem (ERLP) is defined as follows: Inputs to the ERLP consist of a positive integer k and a graph G. The objective of the ERLP is to determine whether there exists an Eulerian circuit, that is, a circuit that contains all of the edges of G, that has no subcycle of length less than k. If G has an Eulerian circuit that has no subcycle of length less than k, and every Eulerian circuit of G has an subcycle of length less than or equal to k, then k is called the Eulerian recurrent length of G, which is abbreviated to the ERL of G, and denoted by ERL(G). In this paper, the ERL’s of complete bipartite graphs are given. Furthermore, upper and lower bounds on the ERL’s of complete graphs are given. Let m and n be positive even integers with m ≥ n. It is shown that ERL(Km_n_ = 2n-4 if n = m ≥ 4, and ERL_Km_n_ = 2n otherwise, where Km_n is the complete bipartite graph whose vertex set V consists of two subsets A containing m vertices and B containing n vertices such that every edge of Km_n connects a vertex in A and one in B. It is also shown that n-4 ≤ ERL_Kn _ ≤ n-2 holds for every odd integer n greater than or equal to 7, where Kn is a complete graph consisting of n vertices.

Original languageEnglish
Pages (from-to)2328-2333
Number of pages6
JournalAdvanced Science Letters
Volume20
Issue number10-12
DOIs
Publication statusPublished - 2014

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Complete Bipartite Graph
Complete Graph
Networks (circuits)
Integer
Less than or equal to
Vertex of a graph
Upper and Lower Bounds
Odd
Subset
Graph in graph theory

Keywords

  • Complete bipartite graph
  • Complete graph
  • Eulerian circuit
  • Graph theory
  • Hamiltonian decomposition
  • Optimization problem

ASJC Scopus subject areas

  • Engineering(all)
  • Environmental Science(all)
  • Computer Science(all)
  • Energy(all)
  • Mathematics(all)
  • Health(social science)
  • Education

Cite this

The quest for the eulerian recurrent lengths of complete bipartite graphs and complete graphs. / Jinbo, Shuji.

In: Advanced Science Letters, Vol. 20, No. 10-12, 2014, p. 2328-2333.

Research output: Contribution to journalArticle

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