For a connected graph G = (V,E) of order at least three, the monophonic distance dm(u, v) is the length of a longest u− v monophonic path in G. For subsets A and B of V , the monophonic distance dm(A,B) is defined as dm(A,B) = min{dm(x, y) : x ∈ A, y ∈ B}. A u− v path of length dm(A,B) is called an A−B detour monophonic path joining the sets A,B ⊆ V, where u ∈ A and v ∈ B. A set S ⊆ E is called...

A many-to-many k-disjoint path cover of a graph joining two disjoint vertex sets S and T of equal size k is a set of k vertex-disjoint paths between S and T that altogether cover every vertex of the graph. The many-to-many k-disjoint path cover is classified as paired if each source in S is further required to be paired with a specific sink in T , or unpaired otherwise. In this paper, we first ...

Notation. (u, v)G is the shortest path between u and v with u, v ∈ V (G) and (u, v)G ⊆ E(G). ‖(u, v)G‖ denotes the length of this path. We will omit the subscript when the context is obvious. Most of the time, we use paths as sets of edges. However, we will occasionally refer to the path’s vertex set; by that, we mean the set of vertices that are endpoints of members of the path. The ends (or e...

Corrádi and Hajnal [1] showed that any graph of order at least 3k with minimum degree at least 2k contains k vertex-disjoint cycles. This minimum degree condition is sharp, because the complete bipartite graph K2k−1,n−2k+1 does not contain k vertex-disjoint cycles. About the existence of vertex-disjoint cycles of the same length, Thomassen [4] conjectured that the same minimum degree condition ...

We prove that if the vertices of a complete graph are labeled with the elements of an arithmetic progression, then for any given vertex there is a Hamiltonian path starting at this vertex such that the absolute values of the differences of consecutive vertices along the path are pairwise distinct. In another extreme case where the label set has small additive energy, we show that the graph actu...

We say that a 3-uniform hypergraph has a Hamilton cycle if there is a cyclic ordering of its vertices such that every pair of consecutive vertices lies in a hyperedge which consists of three consecutive vertices. Also, let C4 denote the 3-uniform hypergraph on 4 vertices which contains 2 edges. We prove that for every ε > 0 there is an n0 such that for every n n0 the following holds: Every 3-un...

The matching graph M(G) of a graph G has a vertex set of all perfect matchings of G, with two vertices being adjacent whenever the union of the corresponding perfect matchings forms a Hamiltonian cycle. We prove that the matching graph M(Qd) of the d-dimensional hypercube is bipartite and connected for d ≥ 4. This proves Kreweras’ conjecture [2] that the graph Md is connected, where Md is obtai...

We determine the maximum number of induced cycles that can be contained in a graph on n ≥ n0 vertices, and show that there is a unique graph that achieves this maximum. This answers a question of Chvátal and Tuza from the 1980s. We also determine the maximum number of odd or even induced cycles that can be contained in a graph on n ≥ n0 vertices and characterise the extremal graphs. This resolv...

We show that in any graph G on n vertices with d(x) + d(y) ≥ n for any two nonadjacent vertices x and y, we can fix the order of k vertices on a given cycle and find a hamiltonian cycle encountering these vertices in the same order, as long as k < n/12 and G is d(k + 1)/2e-connected. Further we show that every b3k/2cconnected graph on n vertices with d(x) + d(y) ≥ n for any two nonadjacent vert...

It is shown that every connected vertex-transitive graph of order 4p, where p is a prime, is hamiltonian with the exception of the Coxeter graph which is known to possess a Hamilton path. In 1969, Lovász [22] asked if every finite, connected vertex-transitive graph has a Hamilton path, that is, a path going through all vertices of the graph. With the exception of K 2 , only four connected verte...