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Tur\'an Density of $2$-edge-colored Bipartite Graphs with Application on $\{2, 3\}$-Hypergraphs
- Publication Year :
- 2020
-
Abstract
- We consider the Tur\'an 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 Tur\'an 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 $h_n(G_n)=(|E_r(G)|+|E_b(G)|)/{n\choose 2}$ is the formula to measure the edge density of $G_n$. We will determine the Tur\'an densities of all $2$-edge-colored bipartite graphs. We also give an important application of our study on the Tur\'an problems of $\{2, 3\}$-hypergraphs.<br />Comment: 21 pages
- Subjects :
- Mathematics - Combinatorics
05C35, 05C65
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2012.06327
- Document Type :
- Working Paper