Research Article
New Sufficient Conditions for Hamiltonian Paths
M. Sohel Rahman,
1
M. Kaykobad,
2
and Jesun Sahariar Firoz
1
1
AℓEDA Group, Department of CSE, BUET, Dhaka 1000, Bangladesh
2
Department of CSE, BUET, Dhaka 1000, Bangladesh
Correspondence should be addressed to M. Sohel Rahman; msrahman@cse.buet.ac.bd
Received 5 March 2014; Revised 2 June 2014; Accepted 3 June 2014; Published 19 June 2014
Academic Editor: Jos´ e M. Sigarreta
Copyright © 2014 M. Sohel Rahman et al. Tis is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
A Hamiltonian path in a graph is a path involving all the vertices of the graph. In this paper, we revisit the famous Hamiltonian
path problem and present new sufcient conditions for the existence of a Hamiltonian path in a graph.
1. Introduction
Hamiltonian paths and cycles are named afer William Rowan
Hamilton who invented the puzzle that involves fnding a
Hamiltonian cycle in the edge graph of the dodecahedron.
Although Hamilton solved this particular puzzle, fnding
Hamiltonian cycles or paths in arbitrary graphs is proved to
be among the hardest problems of computer science [1]. As a
result, instead of complete characterization, most researchers
aimed to fnd sufcient conditions for a graph to possess a
Hamiltonian cycle or path. In this paper, we focus on degree
based sufcient conditions for the existence of Hamiltonian
paths in a graph.
To the best of our knowledge, the quest for good sufcient
degree based conditions for Hamiltonian cycles or paths dates
back to 1952 when Dirac presented the following theorem,
where () denotes the degree of the minimum degree vertex
of the graph .
Teorem 1 (see [2]). If is a simple graph with vertices,
where ≥3 and () ≥ /2, then contains a Hamiltonian
cycle.
Later Ore in 1960 presented a highly celebrated result
where a lower bound for the degree sum of nonadjacent pairs
of vertices was used to force the existence of a Hamiltonian
cycle. In particular, Ore proved the following theorem, where
denotes the degree of the vertex .
Teorem 2 (see [3]). Let be a simple graph with vertices
and , V distinct nonadjacent vertices of with
+
V
≥.
Ten, has a Hamiltonian cycle.
A graph satisfying Ore’s condition has a diameter of
only two [4], where the diameter of a graph is the longest
distance between two vertices. But if a sufcient condition
can be derived for a graph with diameter more than two,
Hamiltonian path or cycle may be found with fewer edges.
With this motivation, Rahman and Kaykobad [5] proposed
a sufcient condition to fnd a Hamiltonian Path in a graph
involving the parameter (, V), which denotes the length of
the shortest path between and V.
Teorem 3 (see [5]). Let = (, ) be a connected graph with
vertices such that for all pairs of distinct nonadjacent vertices
, V ∈ one has
+
V
+ (, V)≥+1. Ten, has a
Hamiltonian path.
In some subsequent literature, the condition “
+
V
+
(, V) ≥ +1, where , V are distinct nonadjacent
vertices of a graph having vertices,” is referred to as the
“Rahman-Kaykobad” condition. A number of interesting
results were achieved extending and using the “Rahman-
Kaykobad” condition as listed below.
Teorem 4 (see [6]). Let be a 2-connected graph which
satisfes the “Rahman-Kaykobad” condition. If contains
Hindawi Publishing Corporation
e Scientific World Journal
Volume 2014, Article ID 743431, 6 pages
http://dx.doi.org/10.1155/2014/743431