Doubly Chorded Cycles in Graphs

In 1963, Corradi and Hajnal proved that for any positive integer k if a graph contains at least 3k vertices and has minimum degree at least 2k, then it contains k disjoint cycles. This result is sharp, meaning there are graphs on at least 3k vertices with a minimum degree of 2k-1 that do not contain k disjoint cycles. Their work is the motivation behind finding sharp conditions that guarantee the existence of specific structures, e.g. cycles, chorded cycles, theta graphs, etc. In this talk, we will explore minimum degree conditions which guarantee a specific number of doubly chorded cycles, graphs that contain a spanning cycle and at least two additional edges, called chords. In particular, we will discuss our findings on these conditions and how it fits in with previous results in this area

Minor-Obstructions for Apex-Pseudoforests

A graph is called a pseudoforest if none of its connected components contains more than one cycle. A graph is an apex-pseudoforest if it can become a pseudoforest by removing one of its vertices. We identify 33 graphs that form the minor-obstruction set of the class of apex-pseudoforests, i.e., the set of all minor-minimal graphs that are not apex-pseudoforests

Cycles are strongly Ramsey-unsaturated

We call a graph H Ramsey-unsaturated if there is an edge in the complement of H such that the Ramsey number r(H) of H does not change upon adding it to H. This notion was introduced by Balister, Lehel and Schelp who also proved that cycles (except for $C_4$) are Ramsey-unsaturated, and conjectured that, moreover, one may add any chord without changing the Ramsey number of the cycle $C_n$, unless n is even and adding the chord creates an odd cycle. We prove this conjecture for large cycles by showing a stronger statement: If a graph H is obtained by adding a linear number of chords to a cycle $C_n$, then $r(H)=r(C_n)$, as long as the maximum degree of H is bounded, H is either bipartite (for even n) or almost bipartite (for odd n), and n is large. This motivates us to call cycles strongly Ramsey-unsaturated. Our proof uses the regularity method

On the homology of locally finite graphs

We show that the topological cycle space of a locally finite graph is a canonical quotient of the first singular homology group of its Freudenthal compactification, and we characterize the graphs for which the two coincide. We construct a new singular-type homology for non-compact spaces with ends, which in dimension~1 captures precisely the topological cycle space of graphs but works in any dimension.Comment: 30 pages. This is an extended version of the paper "The homology of a locally finite graph with ends" (to appear in Combinatorica) by the same authors. It differs from that paper only in that it offers proofs for Lemmas 3, 4 and 10, as well as a new footnote in Section

On Directed Feedback Vertex Set parameterized by treewidth

We study the Directed Feedback Vertex Set problem parameterized by the treewidth of the input graph. We prove that unless the Exponential Time Hypothesis fails, the problem cannot be solved in time $2^{o(t\log t)}\cdot n^{\mathcal{O}(1)}$ on general directed graphs, where $t$ is the treewidth of the underlying undirected graph. This is matched by a dynamic programming algorithm with running time $2^{\mathcal{O}(t\log t)}\cdot n^{\mathcal{O}(1)}$. On the other hand, we show that if the input digraph is planar, then the running time can be improved to $2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}$.Comment: 20