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 language | English |
|---|---|
| Pages (from-to) | 2328-2333 |
| Number of pages | 6 |
| Journal | Advanced Science Letters |
| Volume | 20 |
| Issue number | 10-12 |
| DOIs | |
| Publication status | Published - 2014 |
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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 journal › Article
}
TY - JOUR
T1 - The quest for the eulerian recurrent lengths of complete bipartite graphs and complete graphs
AU - Jinbo, Shuji
PY - 2014
Y1 - 2014
N2 - 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.
AB - 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.
KW - Complete bipartite graph
KW - Complete graph
KW - Eulerian circuit
KW - Graph theory
KW - Hamiltonian decomposition
KW - Optimization problem
UR - http://www.scopus.com/inward/record.url?scp=84919392897&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84919392897&partnerID=8YFLogxK
U2 - 10.1166/asl.2014.5723
DO - 10.1166/asl.2014.5723
M3 - Article
AN - SCOPUS:84919392897
VL - 20
SP - 2328
EP - 2333
JO - Advanced Science Letters
JF - Advanced Science Letters
SN - 1936-6612
IS - 10-12
ER -