Trees Contained in Every Orientation of a Graph
Copyright policy )Trees Contained in Every Orientation of a Graph
For every graph $G$, let $t(G)$ denote the largest integer $t$ such that every oriented tree of order $t$ appears in every orientation of $G$. In 1980, Burr conjectured that $t(G)\geq 1+\chi(G)/2$ for all $G$, and showed that $t(G)\geq 1+ \lfloor\sqrt{\chi(G)}\rfloor$; this bound remains the state of the art, apart from the multiplicative constant. We present an elementary argument that improves this bound whenever $G$ has somewhat large chromatic number, showing that $t(G)\geq \lfloor \chi(G)/\log_2 v(G)\rfloor$ for all $G$.
- Universidade de São Paulo Brazil
Economics, Multiplicative function, Directed graphs (digraphs), tournaments, Geometry, Graph Labeling, Limits and Structures in Graph Theory, Mathematical analysis, Graph, Trees, Upper and lower bounds, oriented graphs, Graph Limits, FOS: Mathematics, Discrete Mathematics and Combinatorics, Graph Labeling and Dimension Problems, Order (exchange), Orientation (vector space), Discrete mathematics, Computer science, anti-dominating set, Programming language, Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.), Computational Theory and Mathematics, Combinatorics, Computer Science, Physical Sciences, Chromatic scale, Simple graph, Integer (computer science), Mathematics, Finance, Graph Theory and Algorithms, Constant (computer programming)
Economics, Multiplicative function, Directed graphs (digraphs), tournaments, Geometry, Graph Labeling, Limits and Structures in Graph Theory, Mathematical analysis, Graph, Trees, Upper and lower bounds, oriented graphs, Graph Limits, FOS: Mathematics, Discrete Mathematics and Combinatorics, Graph Labeling and Dimension Problems, Order (exchange), Orientation (vector space), Discrete mathematics, Computer science, anti-dominating set, Programming language, Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.), Computational Theory and Mathematics, Combinatorics, Computer Science, Physical Sciences, Chromatic scale, Simple graph, Integer (computer science), Mathematics, Finance, Graph Theory and Algorithms, Constant (computer programming)
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