Turán Density of $2$-edge-colored Bipartite Graphs with Application on $\{2,3\}$-Hypergraphs

Abstract

We consider the Turán problems of $2$-edge-colored graphs. A $2$-edge-colored graph $H=(V,E_r,E_b)$ is a triple consisting of the vertex set $V$, the set of red edges $E_r$ and the set of blue edges $E_b$ with $E_r$ and $E_b$ do not have to be disjoint. The Turán density $\pi (H)$ of $H$ is defined to be ${\lim }_{n\to \infty }{\max }_{G_n}{h}_n(G_n)$, where $G_n$ is chosen among all possible $2$-edge-colored graphs on $n$ vertices containing no $H$ as a sub-graph and $h_n(G_n)=(|E_r(G)|+|E_b(G)|)/\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ is the formula to measure the edge density of $G_n$. We will determine the Turán densities of all $2$-edge-colored bipartite graphs. We also give an important application of our study on the Turán problems of $\{2,3\}$-hypergraphs.

21 pages