For a graphG, let diff(G) = p(G)− c(G), where p(G) and c(G) denote the orders of a longest path and a longest cycle inG, respectively. LetG be a 3-connected graph of order n. In the paper, we give a best-possible lower bound to σ4(G) to assure diff(G) ≤ 1. The result settles a conjecture in J. Graph Theory 37 (2001), 137–156.

In this article the Loewy length of the descent algebra of D2m+1 is shown to be m + 2, for m ≥ 2, by providing an upper bound that agrees with the lower bound in [Bonnafé and Pfeiffer, 2006]. The bound is obtained by showing that the length of the longest path in the quiver of the descent algebra of D2m+1 is at most m+1. To achieve this bound, the geometric approach to the descent algebra is us...

Let τ(G) denote the number of vertices in a longest path of the graph G and let k1 and k2 be positive integers such that τ(G) = k1+k2. The question at hand is whether the vertex set V (G) can be partitioned into two subsets V1 and V2 such that τ(G[V1]) ≤ k1 and τ(G[V2]) ≤ k2. We show that several classes of graphs have this partition property.

In this paper, we study triangle-free graphs. Let G = (VG, EG) be an arbitrary triangle-free graph with minimum degree at least two and σ4(G) ≥ |VG|+2. We first show that either for any path P in G there exists a cycle C such that |VP \ VC | ≤ 1, or G is isomorphic to exactly one exception. Using this result, we show that for any set S of at most δ vertices in G there exists a cycle C such that...

Yasir BASHIR1,∗, Faisal NADEEM, Ayesha SHABBIR Faculty of Mathematics, COMSATS Institute of Information Technology Wah Campus, Wah Cantt, Pakistan Faculty of Mathematics, COMSATS Institute of Information Technology Lahore Campus, Lahore, Pakistan Abdus Salam School of Mathematical Sciences, GC University, 68-B, New Muslim Town, Lahore, Pakistan University College of Engineering, Sciences and Te...

In this note, we show that some problems related to the length of the longest simple path from a given vertex in a graph are NP-complete. We also discuss an extension to the graph coloring problem. @ 2004 Elsevier Ltd. All rights reserved. Keywords-Longest path, Graph coloring, Complexity.

The path length of a tree, that is the sum of the lengths of all root-leaf paths, is an important measure of eeciency of the tree. The fringe of a tree is the diierence between the lengths of the longest path and the shortest path in the tree. The minimal and the maximal path length of trees with N leaves and given fringe have been well studied. In this paper, we initiate the study of the expec...

We consider the problem of finding a long, simple path in an undirected graph. We present a polynomial-time algorithm that finds a path of length Ω ( (logL/ log logL)2 ) , where L denotes the length of the longest simple path in the graph. This establishes the performance ratio O ( n(log logn/ logn)2 ) for the longest path problem, where n denotes the number of vertices in the graph.