KAIST Discrete Math Seminar

A chordless cycle in a graph G is an induced subgraph of G which is a cycle of length at least four. We prove that the Erdős-Pósa property holds for chordless cycles, which resolves the major open question concerning the Erdős-Pósa property. Our proof for chordless cycles is constructive: in polynomial time, one can find either k+1 vertex-disjoint chordless cycles, or ck² log k vertices hitting every chordless cycle for some constant c. It immediately implies an approximation algorithm of factor O(OPT log OPT) for Chordal Vertex Deletion. We complement our main result by showing that chordless cycles of length at least ℓ for any fixed ℓ≥ 5 do not have the Erdős-Pósa property.

We prove that a class of graphs obtained by gluing complete multipartite graphs in a tree-like way satisfies a conjecture of Kohayakawa, Nagle, Rödl, and Schacht on random-like counts for small graphs in locally dense graphs. This implies an approximate version of the conjecture for graphs with bounded tree-width. We also prove an analogous result for odd cycles instead of complete multipartite graphs.
The proof uses a general information theoretic method to prove graph homomorphism inequalities for tree-like structured graphs, which may be of independent interest.

A classical result of Komlós, Sárközy and Szemerédi states that every n-vertex graph with minimum degree at least (1/2 +o(1))n contains every n-vertex tree with maximum degree at most O(n/log n) as a subgraph, and the bounds on the degree conditions are sharp.
On the other hand, Krivelevich, Kwan and Sudakov recently proved that for every n-vertex graph G with minimum degree at least αn for any fixed α>0 and every n-vertex tree T with bounded maximum degree, one can still find a copy of T in G with high probability after adding O(n) randomly-chosen edges to G.
We extend this result to trees with unbounded maximum degree. More precisely, for a given n^ε ≤ Δ≤ cn/log n and α>0, we determined the precise number (up to a constant factor) of random edges that we need to add to an arbitrary n-vertex graph G with minimum degree αn in order to guarantee with high probability a copy of any fixed T with maximum degree at most Δ. This is joint work with Felix Joos.

Let f(n,r) denote the maximum number of colourings of A⊆{1,…,n} with r colours such that each colour class is sum-free. Here, a sum is a subset {x,y,z} such that x+y=z. We show that f(n,2) = 2^⌈n/2⌉, and describe the extremal subsets. Further, using linear optimisation, we asymptotically determine the logarithm of f(n,r) for r≤5.
Joint work with Maryam Sharifzadeh and Katherine Staden.

Bootstrap percolation is a simple process which is used as a model of propagation phenomena in real world networks including, for example, the spread of a rumour in a social network, the dynamics of ferromagnetism and information processing in neural networks. Given a graph G and an integer r, the r-neighbour bootstrap process begins with a set of “infected” vertices and, in each step, a “healthy” vertex becomes infected if it has at least r infected neighbours. A central problem in the area is to determine the size of the smallest initial infection which will spread to every vertex of the graph. In this talk, I will present a trick for obtaining lower bounds on this quantity by transforming the problem into an infection problem on the edges of the graph and applying some basic facts from linear algebra. In particular, I will outline a proof of a conjecture of Balogh and Bollobás (2006) on the smallest infection which spreads to every vertex of a high-dimensional square lattice and mention some potential applications to analysing the behaviour of a random infection in this setting. This talk is based on joint work with Natasha Morrison.