Acta Mathematicae Applicatae Sinica, English Series
Vol. 33, No. 3 (2017) 789–798
http://www.ApplMath.com.cn & www.SpringerLink.com
Acta MathemaƟcae Applicatae Sinica,
The Editorial Office of AMAS &
Springer-Verlag Berlin Heidelberg 2017
On Heavy Paths in 2-connected Weighted Graphs
Bin-long LI, Sheng-gui ZHANG
Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi’an 710072,
Abstract A weighted graph is a graph in which every edge is assigned a non-negative real number. In a
weighted graph, the weight of a path is the sum of the weights of its edges, and the weighed degree of a vertex is
the sum of the weights of the edges incident with it. In this paper we give three weighted degree conditions for
the existence of heavy or Hamilton paths with one or two given end-vertices in 2-connected weighted graphs.
Keywords weighted graph; heavy path; weighed degree
2000 MR Subject Classiﬁcation 05C22; 05C38
We use Bondy and Murty
for terminology and notation not deﬁned here and consider ﬁnite
simple graphs only.
A weighted graph is a graph in which every edge e is assigned a non-negative real number
w(e), called the weight of e.LetG =(V,E) be a weighted graph. For a subgraph H of G,
V (H)andE(H) denote the sets of vertices and edges of H, respectively. The weight of H is
For a vertex v ∈ V , N
(v) denotes the set, and d
(v) the number, of vertices in H that are
adjacent to v. We deﬁne the weighted degree of v in H by
When no confusion occurs, we will denote N
An unweighted graph can be regarded as a weighted graph in which each edge e is assigned
weight w(e) = 1. Thus, in an unweighted graph, d
(v)=d(v) for every vertex v,andthe
weight of a subgraph is simply the number of its edges.
In 1989, Bondy and Fan
began the study of the existence of heavy paths and cycles in
weighted graphs. They showed two results with Dirac-type weighted degree condition. In the
following by an x-path we mean a path whose initial vertex is x;andbyan(x, y)-path we mean
one whose end-vertices are x and y.
Manuscript received December 30, 2010. Revised June 23, 2017.
Supported by the National Natural Science Foundation of China (No. 11571135, 11601429 and 11671320) and
the Natural Science Foundation of Shaanxi Province (No. 2016JQ1002).