We prove that every simplicial automorphism of the free splitting graph of a free group Fn is induced by an outer automorphism of Fn for n ≥ 3. In this note we consider the graph Gn of free splittings of the free group Fn of rank n ≥ 3. Loosely speaking, Gn is the graph whose vertices are non-trivial free splittings of Fn up to conjugacy, and where two vertices are adjacent if they are represen...

A unichord in a graph is an edge that is the unique chord of a cycle. A square is an induced cycle on four vertices. A graph is unichord-free if none of its edges is a unichord. We give a slight restatement of a known structure theorem for unichord-free graphs and use it to show that, with the only exception of the complete graph K4, every square-free, unichord-free graph of maximum degree 3 ca...

A graph is claw-free if it does not contain K I ,3 as an induced subgraph. A graph is KI,r-free if it does not contain KI,r as an induced subgraph. In this paper, we find bounds on the minimum number of edges needed to ensure a KI,.,.-free contains k vertex disjoint cycles. The bound on claw-free graphs is sharp.

We introduce the convex circuit-free coloration and convex circuit-free chromatic number −→ χa( −→ G) of an oriented graph −→ G and establish various basic results. We show that the problem of deciding if an oriented graph verifies χa( −→ G ) ≤ k is NP-complete if k ≥ 3 and polynomial if k ≤ 2. We exhibit an algorithm which finds the optimal convex circuit-free coloration for tournaments, and c...

In this paper we investigate the ratio of the connected vertex cover number to the vertex cover number in graphs, called the Price of Connectivity (PoC). For general graphs, this ratio is strictly bounded by 2. We prove that for every (P5, C5, C4)-free graph the ratio equals 1. We prove also that for every (P5, C4)-free graph the ratio is bounded by 4/3 and that for every (P7, C6,∆1,∆2)-free gr...

We show that every K4-free graph G with n vertices can be made bipartite by deleting at most n/9 edges. Moreover, the only extremal graph which requires deletion of that many edges is a complete 3-partite graph with parts of size n/3. This proves an old conjecture of P. Erdős.

The toughness of a non-complete graph G is the minimum value of |S| ω(G−S) among all separating vertex sets S ⊂ V (G), where ω(G − S) > 2 is the number of components of G−S. It is well-known that every 3-connected planar graph has toughness greater than 1/2. Related to this property, every 3-connected planar graph has many good substructures, such as a spanning tree with maximum degree three, a...

graph theory based methods are powerful means for representing structural systems so that their geometry and topology can be understood clearly. the combination of graph theory based methods and some metaheuristics can offer effective solutions for complex engineering optimization problems. this paper presents a charged system search (css) algorithm for the free shape optimizations of thin-wall...

In this paper, we show that every 3-connected claw-free graph G has a 2-factor having at most max { 2 5(α+1), 1 } cycles, where α is the independence number of G. As a corollary of this result, we also prove that every 3-connected claw-free graph G has a 2-factor with at most ( 4|G| 5(δ+2)+ 2 5 ) cycles, where δ is the minimum degree of G. This is an extension of a known result on the number of...

Let X, Y be connected graphs. A graph G is (X, Y )-free if G contains a copy of neither X nor Y as an induced subgraph. Pairs of connected graphs X, Y such that every 3-connected (X, Y )-free graph is Hamilton connected have been investigated most recently in [8] and [5]. This paper improves those results. Specifically, it is shown that every 3-connected (X, Y )-free graph is Hamilton connected...