Główne pojęcia
The author aims to reduce Linear Hadwiger's Conjecture to graph coloring by proving that every graph with no Kt minor is O(t log log t)-colorable, using innovative approaches.
Streszczenie
The content discusses reducing Linear Hadwiger's Conjecture to coloring small graphs. It presents key theorems and proofs related to graph theory, connectivity, and chromatic numbers. The author introduces novel methods for building minors in graphs and establishes corollaries based on existing conjectures.
The main focus is on proving that Kt-minor-free graphs are O(t log log t)-colorable, providing insights into the complexity of graph coloring problems.
Key results include bounds on chromatic numbers, density theorems for graphs, and strategies for constructing highly-connected subgraphs in Kt-minor-free graphs.
Statystyki
In 1943, Hadwiger conjectured that every graph with no Kt minor is (t − 1)-colorable.
Every graph with no Kt minor has average degree O(t√log t) and hence is O(t√log t)-colorable.
Recent research showed that every graph with no Kt minor is O(t(log t)β)-colorable for every β > 1/4.
The first main result of the paper is that every graph with no Kt minor is O(t log log t)-colorable.
For every integer r ≥ 3, there exists tr such that for all integers t ≥ tr, every Kr-free Kt-minor-free graph is Ct-colorable.
Cytaty
"Every graph with no Kt minor has average degree O(t√log t) and hence is O(t√log t)-colorable." - Author
"Linear Hadwiger’s Conjecture holds if the clique number of the graph is small as a function of t." - Author
"The proof presented here is independent of previous work." - Author