ON CHROMATIC UNIQUENESS OF SOME COMPLETE TRIPARTITE GRAPHS

Let \(P(G, x)\) be a chromatic polynomial of a graph \(G\). Two graphs \(G\) and \(H\) are called chromatically equivalent iff \(P(G, x) = H(G, x)\). A graph \(G\) is called chromatically unique if \(G\simeq H\) for every \(H\) chromatically equivalent to \(G\). In this paper, the chromatic uniqueness of complete tripartite graphs \(K(n_1, n_2, n_3)\) is proved for \(n_1 \geqslant n_2 \geqslant n_3 \geqslant 2\) and \(n_1 - n_3 \leqslant 5\).

User opinions driven social recommendation system

Recommendation systems provide reliable and relevant recommendations to users and also enable users’ trust on the website. This is achieved by the opinions derived from reviews, feedbacks and preferences provided by the users when the product is purchased or viewed through social networks. This integrates interactions of social networks with recommendation systems which results in the behavior of users and user’s friends. The techniques used so far for recommendation systems are traditional, based on collaborative filtering and content based filtering. This paper provides a novel approach called User-Opinion-Rating (UOR) for building recommendation systems by taking user generated opinions over social networks as a dimension. Two tripartite graphs namely User-Item-Rating and User-Item-Opinion are constructed based on users’ opinion on items along with their ratings. Proposed approach quantifies the opinions of users and results obtained reveal the feasibility.

Triangle-degrees in graphs and tetrahedron coverings in 3-graphs

We investigate a covering problem in 3-uniform hypergraphs (3-graphs): Given a 3-graph F, what is c1(n, F), the least integer d such that if G is an n-vertex 3-graph with minimum vertex-degree $\delta_1(G)>d$ then every vertex of G is contained in a copy of F in G? We asymptotically determine c1(n, F) when F is the generalized triangle $K_4^{(3)-}$ , and we give close to optimal bounds in the case where F is the tetrahedron $K_4^{(3)}$ (the complete 3-graph on 4 vertices). This latter problem turns out to be a special instance of the following problem for graphs: Given an n-vertex graph G with $m> n^2/4$ edges, what is the largest t such that some vertex in G must be contained in t triangles? We give upper bound constructions for this problem that we conjecture are asymptotically tight. We prove our conjecture for tripartite graphs, and use flag algebra computations to give some evidence of its truth in the general case.