Stephan Brandt

Degree conditions for 2-factors

For any positive integer k, we investigate degree conditions implying that a graph G of order n contains a 2-factor with exactly k components (vertex disjoint cycles). In particular, we prove that for k ≤ (n/4), Ores classical condition for a graph to be hamiltonian (k = 1) implies that the graph contains a 2-factor with exactly k components. We also obtain a sufficient degree condition for a graph to have k vertex disjoint cycles, at least s of which are 3-cycles and the remaining are 4-cycles for any s ≤ k.

Weakly pancyclic graphs

The problem was posed of determining the biclique partition number of the complement of a Hamiltonian path (Monson, Rees, and Pullman, Bull. Inst. Combinatorics and Appl. 14 (1995), 17–86). We define the complement of a path P, denoted

Closure and stable Hamiltonian properties in claw-free graphs

overline{P}

The Erdos-Sós conjecture for graphs of girth 5

, as the complement of P in Km,n where P is a subgraph of Km,n for some m and n. We give an exact formula for the biclique partition number of the complement of a path. In particular, we solve the problem posed in [9]. We also summarize our more general results on biclique partitions of the complement of forests.

A sufficient condition for all short cycles

We consider the equivalence classes of graphs induced by the unsigned versions of the Reidemeister moves on knot diagrams. Any graph that is reducible by some finite sequence of these moves, to a graph with no edges, is called a knot graph. We show that the class of knot graphs strictly contains the set of delta-wye graphs. We prove that the dimension of the intersection of the cycle and cocycle spaces is an effective numerical invariant of these classes.

On the Structure of Dense Triangle-Free Graphs

Abstract We prove that every graph of girth at least 5 with minimum degree δ ⩾ k /2 contains every tree with k edges, whose maximum degree does not exceed the maximum degree of the graph. An immediate consequence is that the famous Erdős-Sos Conjecture, saying that every graph of order n with more than n ( k − 1)/2 edges contains every tree with k edges, is true for graphs of girth at least 5.

Subtrees and subforests of graphs

Abstract Generalizing a result of Haggkvist et al. (1981), we prove that every non-bipartite graph of order n with more than (n − 1) 2 4 + 1 edges contains cycles of every length between 3 and the length of a longest cycle.

Global Connectivity And Expansion: Long Cycles and Factors In f -Connected Graphs

As a consequence of an early result of Pach we show that every maximal triangle-free graph is either homomorphic with a member of a specific infinite sequence of graphs or contains the Petersen graph minus one vertex as a subgraph. From this result and further structural observations we derive that, if a (not necessarily maximal) triangle-free graph of order n has minimum degree δ≥n/3, then the graph is either homomorphic with a member of the indicated family or contains the Petersen graph with one edge contracted. As a corollary we get a recent result due to Chen, Jin and Koh. Finally, we show that every triangle-free graph with δ>n/3 is either homomorphic with C5 or contains the Mobius ladder. A major tool is the observation that every triangle-free graph with δ≥n/3 has a unique maximal triangle-free supergraph.

9-Connected Claw-Free Graphs Are Hamilton-Connected

Abstract In this paper we present sufficient edge-number and degree conditions for a graph to contain all forests of given size. The edge-number bound answers in the affirmative a conjecture due to Erdős and Sos. Furthermore, we will give improved bounds for specified spanning subtrees of graphs.

Triangle-free graphs and forbidden subgraphs

Given a function f : ℕ→ℝ, call an n-vertex graph f-connected if separating off k vertices requires the deletion of at least f(k) vertices whenever k≤(n−f(k))/2. This is a common generalization of vertex connectivity (when f is constant) and expansion (when f is linear). We show that an f-connected graph contains a cycle of length linear in n if f is any linear function, contains a 1-factor and a 2-factor if f(k)≥2k+1, and contains a Hamilton cycle if f(k)≥2(k+1)2. We conjecture that linear growth of f suffices to imply hamiltonicity.