기초과학연구원 이산수학그룹
We prove a conjecture of Boros, Caro, Furedi and Yuster on the maximum number of edges in a 2-connected graph without repeated cycle lengths, which is a restricted version of a longstanding problem of Erdos. Our proof together with the matched lower bound construction of Boros, Caro, Furedi and Yuster show that this problem can …
Specifying a computational problem requires fixing encodings for input and output: encoding graphs as adjacency matrices, characters as integers, integers as bit strings, and vice versa. For such discrete data, the actual encoding is usually straightforward and/or complexity-theoretically inessential (up to polynomial time, say). But concerning continuous data, already real numbers naturally suggest various encodings with very different computational properties. …
This talk considers the following question at the intersection of extremal and structural graph theory: What is the maximum number of copies of a fixed forest $T$ in an $n$-vertex graph in a graph class $\mathcal{G}$ as $n\to \infty$? I will answer this question for a variety of sparse graph classes $\mathcal{G}$. In particular, we show that the answer is …
We prove that if $n \geq 3$, then any family of $3n-3$ sets of matchings of size $n$ in any graph has a rainbow matching of size $n$. This improves on a previous result, in which $3n-3$ is replaced by $3n-2$. We also prove a "cooperative" generalization: for $t > 0$ and $n \geq 3$, …
Erdős and Pósa proved in 1965 that there is a duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold if we restrict to odd cycles. However, in 1999, Reed proved an analogue for odd cycles by relaxing packing …
Graph saturation is one of the oldest areas of investigation in extremal combinatorics. A graph $G$ is called $F$-saturated if $G$ does not contain a subgraph isomorphic to $F$, but the addition of any edge creates a copy of $F$. The function $\operatorname{sat}(n,F)$ is defined to be the minimum number of edges in an $n$-vertex …
We propose a theoretical analysis of recommendation systems in an online setting, where items are sequentially recommended to users over time. In each round, a user, randomly picked from a population of m users, requests a recommendation. The decision-maker observes the user and selects an item from a catalogue of n items. Importantly, an item …
Corneil, Olariu, and Stewart presented a recognition algorithm for interval graphs by six graph searches. Li and Wu simplified it to only four. The great simplicity of the latter algorithm is however eclipsed by the complicated and long proofs. The main purpose of this paper is to present a new and significantly shorter proof for …
In 1975, Erdős asked the following question: what is the smallest function $f(n)$ for which all graphs with $n$ vertices and $f(n)$ edges contain two edge-disjoint cycles $C_1$ and $C_2$, such that the vertex set of $C_2$ is a subset of the vertex set of $C_1$ and their cyclic orderings of the vertices respect each …
The twin-width of a graph G can be defined as the least integer d such that there is a sequence of length |V(G)| of (strictly) coarser and coarser partitions of its vertex set V(G), and every part X of every partition P of the sequence has at most d other parts Y of P with …