Let G be a graph with chromatic number χ, maximum degree ∆ and clique number ω. Reed's conjecture states that χ ≤ ⌈(1 − ε)(∆ + 1) + εω⌉ for all ε ≤ 1/2. It was shown by King and Reed that, provided ∆ is large enough, the conjecture holds for ε ≤ 1/130, 000. In this article, we show that the same statement holds for ε ≤ 1/26, thus making a significant step towards Reed's conjecture. We derive this result from a general technique to bound the chromatic number of a graph where no vertex has many edges in its neighbourhood. Our improvements to this method also lead to improved bounds on the strong chromatic index of general graphs. We prove that χ ′ s (G) ≤ 1.835∆(G) 2 provided ∆(G) is large enough.
Gallai's path decomposition conjecture states that the edges of any connected graph on n vertices can be decomposed into at most n+1 2 paths. We confirm that conjecture for all graphs with maximum degree at most five.
We prove that the roots of the chromatic polynomials of planar graphs are dense in the interval between 32/27 and 4, except possibly in a small interval around τ + 2 where τ is the golden ratio. This interval arises due to a classical result of Tutte, which states that the chromatic polynomial of every planar graph takes a positive value at τ + 2. Our results lead us to conjecture that τ + 2 is the only such number less than 4.
The Tutte polynomial of a graph is a two-variable polynomial whose zeros and evaluations encode many interesting properties of the graph. In this article we investigate the zeros of the Tutte polynomials of graphs, and show that they form a dense subset of certain regions of the plane. This is the first density result for the zeros of the Tutte polynomial in a region of positive volume. Our result almost confirms a conjecture of Jackson and Sokal except for one region which is related to an open problem on flow polynomials.
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