In this paper, we aim to embed a longest path between every two vertices of a star graph with at most n-3 random edge faults, where n is the dimension of the star graph. Since the star graph is regular of degree n1, n-3 (edge faults) is maximum in the worst case. We show that when n≥6 and there are n-3 edge faults, the star graph can embed a longest path between every two vertices, exclusive of...

We consider the so-called Path Partition Conjecture for digraphs which states that for every digraph, D, and every choice of positive integers, λ1, λ2, such that λ1 + λ2 equals the order of a longest directed path in D, there exists a partition of D into two digraphs, D1 and D2, such that the order of a longest path in Di is at most λi, for i = 1, 2. We prove that certain classes of digraphs, w...

Consider the square [0, n] with points from a Poisson point process of intensity 1 distributed within it. In [4] it was shown that the length Ln of a maximal increasing path (an increasing path that contains the most number of points) when properly centered and scaled converges to the Tracy-Widom distribution. Later [16] showed that all maximal paths have a displacement of n 2 3 +o(1) from the ...

Andreas Björklund and Thore Husfeldt Department of Computer S ien e, Lund University Abstra t. We onsider the problem of nding a long, simple path in an undire ted graph. We present a polynomial-time algorithm that nds a path of length Ω (logL/ log logL) , where L denotes the length of the longest simple path in the graph. This establishes the performan e ratio O |V |(log log |V |/ log |V |) fo...

The Path Partition Conjecture for digraphs states that for every digraph D, and every choice of positive integers λ1, λ2 such that λ1 + λ2 equals the order of a longest directed path in D, there exists a partition of D in two subdigraphs D1,D2 such that the order of the longest path in Di is at most λi for i = 1, 2. We present sufficient conditions for a digraph to satisfy the Path Partition Co...

Let G be a connected graph on n vertices. The Gallai number Gal(G) of is the size smallest set vertices that meets every maximum path in G. Grünbaum constructed with Gal(G)=3. Very recently, Long, Milans, and Munaro, proved Gal(G)≤8n3/4. This was first sub-linear upper bound terms n. We improve their to Gal(G)≤5n2/3. also tighten more general result Long et al. For multigraph M, we prove if L(M...

The snake-in-the-box problem is concerned with finding a longest induced path in a hypercube Qn. Similarly, the coil-in-the-box problem is concerned with finding a longest induced cycle in Qn. We consider a generalization of these problems that considers paths and cycles where each pair of vertices at distance at least k in the path or cycle are also at distance at least k in Qn. We call these ...

The longest path problem is the problem of finding a simple path of maximum length in a graph. Polynomial solutions for this problem are known only for special classes of graphs, while it is NP-hard on general graphs. In this paper we are proposing a O(n) time algorithm to find the longest path on biconvex graphs, where n is the number of vertices of the input graph. We have used Dynamic Progra...