arXiv subjects
Publications and source records attributed to He Guo.
In this paper we solve two problems of Esperet, Kang and Thomasse as well as Li concerning (i) induced bipartite subgraphs in triangle-free graphs and (ii) van der Waerden numbers. Each time random greedy algorithms allow us to go beyond the Lovasz Local Lemma or alteration method used in previous work, illustrating the power of the algorithmic approach to the probabilistic method.
The Prague dimension of graphs was introduced by Nesetril, Pultr and Rodl in the 1970s. Proving a conjecture of Furedi and Kantor, we show that the Prague dimension of the binomial random graph is typically of order n/log n for constant edge-probabilities. The main new proof ingredient is a Pippenger-Spencer type edge-coloring result for random hypergraphs with large uniformities, i.e., edges of size O(log n).
A network is a typical expressive form of representing complex systems in terms of vertices and links, in which the pattern of interactions amongst components of the network is intricate. The network can be static that does not change over time or dynamic that evolves through time. The complication of network analysis is different under the new circumstance of network size explosive increasing. In this paper, we introduce a new network science concept called big network. Big networks are generally in large-scale with a complicated and higher-order inner structure. This paper proposes a guideline framework that gives an insight into the major topics in the area of network science from the viewpoint of a big network. We first introduce the structural characteristics of big networks from three levels, which are micro-level, meso-level, and macro-level. We then discuss some state-of-the-art advanced topics of big network analysis. Big network models and related approaches, including ranking methods, partition approaches, as well as network embedding algorithms are systematically introduced. Some typical applications in big networks are then reviewed, such as community detection, link prediction, recommendation, etc. Moreover, we also pinpoint some critical open issues that need to be investigated further.
In 1995 Kim famously proved the Ramsey bound R(3,t) \ge c t^2/\log t by constructing an n-vertex graph that is triangle-free and has independence number at most C \sqrt{n \log n}. We extend this celebrated result, which is best possible up to the value of the constants, by approximately decomposing the complete graph K_n into a packing of such nearly optimal Ramsey R(3,t) graphs. More precisely, for any ε>0 we find an edge-disjoint collection (G_i)_i of n-vertex graphs G_i \subseteq K_n such that (a) each G_i is triangle-free and has independence number at most C_ε\sqrt{n \log n}, and (b) the union of all the G_i contains at least (1-ε)\binom{n}{2} edges. Our algorithmic proof proceeds by sequentially choosing the graphs G_i via a semi-random (i.e., Rodl nibble type) variation of the triangle-free process. As an application, we prove a conjecture in Ramsey theory by Fox, Grinshpun, Liebenau, Person, and Szabo (concerning a Ramsey-type parameter introduced by Burr, Erdos, Lovasz in 1976). Namely, denoting by s_r(H) the smallest minimum degree of r-Ramsey minimal graphs for H, we close the existing logarithmic gap for H=K_3 and establish that s_r(K_3) = Θ(r^2 \log r).