On the minimal length of the longest trail in a fixed edge-density graph

Abstract

A nearly sharp lower bound on the length of the longest trail in a graph on vertices and average degree is given provided the graph is dense enough ( ≥ 12 5). MSC:05C35

Ordering Protein Contact Matrices

“…The assignment problem was formalized in the present work as the resolution of the longest trail problem [ 36 , 42 ]. To the best of our knowledge, this is the first time in the field of structural biology that this problem is formalized in a such way.…”

“…Nevertheless, some results have been already obtained in the literature concerning the longest trail problem. A lower bound on the length of the longest trail was proposed in [36] for enough dense graphs. The author proves that for a connected graph G of n nodes such that its average degree is at least k ≥ 12.5, there exists a trail of length at least or n ≤⌈ k ⌉ + 2.…”

Ramsey numbers of trails and circuits

“…For the lower bound, suppose that we are given a red/blue colouring of the complete graph $${K}_{n}$$. After removing, for each colour class, a suitable forest that meets all odd degree vertices (see, e.g., [3, Proposition 2.1]), we may assume that each colour class is Eulerian, in the sense that every vertex has even degree in both red and blue. However, this is not immediately helpful, since the colour classes may be disconnected.…”

“…After deleting a bounded number of vertices, we may also assume that every vertex has red degree at least $$n\unicode{x02215}2$$, which implies that the remaining graph is connected. By Szécsi [3, Theorem 1.3], which says that any connected graph with average degree $$t$$ contains a trail of length $$\left(\genfrac{}{}{0.0pt}{}{t}{2}\right)+O(t)$$, this then implies that there is a red trail of length at least $$2{n}^{2}\unicode{x02215}9+O(n)$$, as required. Since the component containing this trail is Eulerian, we also have a red circuit of at least the same length.…”