A proof of the four-colour theorem

The four-colour problem remained unsolved for more than a hundred years has played a role of the utmost importance in the development of graph theory. The four-colour theorem was confirmed in 1976, which is not completely satisfied due to: i) part of the proof using computers cannot be verified by hand; ii) even the part, supposedly hand-checkable, is extraordinarily complicated and tedious, and as far as we know, no one has entirely verified it. Seeking a hand-checkable proof of the four-colour theorem is one of world-interested problems, which is addressed in this paper. A necessary and sufficient condition for n-colour theorem in a space is: there exists a largest n-complete graph base in the same space. Examples are given to illustrate applications

Influence Maximization Meets Efficiency and Effectiveness: A Hop-Based Approach

Influence Maximization is an extensively-studied problem that targets at selecting a set of initial seed nodes in the Online Social Networks (OSNs) to spread the influence as widely as possible. However, it remains an open challenge to design fast and accurate algorithms to find solutions in large-scale OSNs. Prior Monte-Carlo-simulation-based methods are slow and not scalable, while other heuristic algorithms do not have any theoretical guarantee and they have been shown to produce poor solutions for quite some cases. In this paper, we propose hop-based algorithms that can easily scale to millions of nodes and billions of edges. Unlike previous heuristics, our proposed hop-based approaches can provide certain theoretical guarantees. Experimental evaluations with real OSN datasets demonstrate the efficiency and effectiveness of our algorithms.Comment: Extended version of the conference paper at ASONAM 2017, 11 page

Towards Profit Maximization for Online Social Network Providers

Online Social Networks (OSNs) attract billions of users to share information and communicate where viral marketing has emerged as a new way to promote the sales of products. An OSN provider is often hired by an advertiser to conduct viral marketing campaigns. The OSN provider generates revenue from the commission paid by the advertiser which is determined by the spread of its product information. Meanwhile, to propagate influence, the activities performed by users such as viewing video ads normally induce diffusion cost to the OSN provider. In this paper, we aim to find a seed set to optimize a new profit metric that combines the benefit of influence spread with the cost of influence propagation for the OSN provider. Under many diffusion models, our profit metric is the difference between two submodular functions which is challenging to optimize as it is neither submodular nor monotone. We design a general two-phase framework to select seeds for profit maximization and develop several bounds to measure the quality of the seed set constructed. Experimental results with real OSN datasets show that our approach can achieve high approximation guarantees and significantly outperform the baseline algorithms, including state-of-the-art influence maximization algorithms.Comment: INFOCOM 2018 (Full version), 12 page

Veech surfaces and simple closed curves

We study the SL(2,R)-infimal lengths of simple closed curves on half-translation surfaces. Our main result is a characterization of Veech surfaces in terms of these lengths. We also revisit the "no small virtual triangles" theorem of Smillie and Weiss and establish the following dichotomy: the virtual triangle area spectrum of a half-translation surface either has a gap above zero or is dense in a neighborhood of zero. These results make use of the auxiliary polygon associated to a curve on a half-translation surface, as introduced by Tang and Webb.Comment: 12 pages. v2: added proof of continuity of infimal length functions on quadratic differential space; 16 pages, one figure; to appear in Israel J. Mat