An algorithmic framework for colouring locally sparse graphs

We develop an algorithmic framework for graph colouring that reduces the problem to verifying a local probabilistic property of the independent sets. With this we give, for any fixed $k\ge 3$ and $\varepsilon>0$, a randomised polynomial-time algorithm for colouring graphs of maximum degree $\Delta$ in which each vertex is contained in at most $t$ copies of a cycle of length $k$, where $1/2\le t\le \Delta^\frac{2\varepsilon}{1+2\varepsilon}/(\log\Delta)^2$, with $\lfloor(1+\varepsilon)\Delta/\log(\Delta/\sqrt t)\rfloor$ colours. This generalises and improves upon several notable results including those of Kim (1995) and Alon, Krivelevich and Sudakov (1999), and more recent ones of Molloy (2019) and Achlioptas, Iliopoulos and Sinclair (2019). This bound on the chromatic number is tight up to an asymptotic factor $2$ and it coincides with a famous algorithmic barrier to colouring random graphs.Comment: 23 page

An improved procedure for colouring graphs of bounded local density

We develop an improved bound for the chromatic number of graphs of maximum degree $\Delta$ under the assumption that the number of edges spanning any neighbourhood is at most $(1-\sigma)\binom{\Delta}{2}$ for some fixed $0<\sigma<1$. The leading term in this bound is best possible as $\sigma\to0$. As two consequences, we advance the state of the art in two longstanding and well-studied graph colouring conjectures, the Erd\H{o}s-Ne\v{s}et\v{r}il conjecture and Reed's conjecture. We prove that the strong chromatic index is at most $1.772\Delta^2$ for any graph $G$ with sufficiently large maximum degree $\Delta$. We prove that the chromatic number is at most $\lceil 0.801(\Delta+1)+0.199\omega\rceil$ for any graph $G$ with clique number $\omega$ and sufficiently large maximum degree $\Delta$.Comment: 21 page

An algorithmic framework for colouring locally sparse graphs

We develop an algorithmic framework for graph colouring that reduces the problem to verifying a local probabilistic property of the independent sets. With this we give, for any fixed k ≥ 3 and ε > 0, a randomised polynomial-time algorithm for colouring graphs of maximum degree ∆ in which each vertex is contained in at most t copies of a cycle of length k, where 1/2 ≤ t ≤ ∆^(2ε / (1+2ε)) / (log ∆) 2 , with (1 + ε)∆/ log(∆/ √ t) colours. This generalises and improves upon several notable results including those of Kim (1995) and Alon, Krivelevich and Sudakov (1999), and more recent ones of Molloy (2019) and Achlioptas, Iliopoulos and Sinclair (2019). This bound on the chromatic number is tight up to an asymptotic factor 2 and it coincides with a famous algorithmic barrier to colouring random graphs

An algorithmic framework for colouring locally sparse graphs

We develop an algorithmic framework for graph colouring that reduces the problem to verifying a local probabilistic property of the independent sets. With this we give, for any fixed k ≥ 3 and ε > 0, a randomised polynomial-time algorithm for colouring graphs of maximum degree ∆ in which each vertex is contained in at most t copies of a cycle of length k, where 1/2 ≤ t ≤ ∆^(2ε / (1+2ε)) / (log ∆) 2 , with (1 + ε)∆/ log(∆/ √ t) colours. This generalises and improves upon several notable results including those of Kim (1995) and Alon, Krivelevich and Sudakov (1999), and more recent ones of Molloy (2019) and Achlioptas, Iliopoulos and Sinclair (2019). This bound on the chromatic number is tight up to an asymptotic factor 2 and it coincides with a famous algorithmic barrier to colouring random graphs

An algorithmic framework for colouring locally sparse graphs

We develop an algorithmic framework for graph colouring that reduces the problem to verifying a local probabilistic property of the independent sets. With this we give, for any fixed k ≥ 3 and ε > 0, a randomised polynomial-time algorithm for colouring graphs of maximum degree ∆ in which each vertex is contained in at most t copies of a cycle of length k, where 1/2 ≤ t ≤ ∆^(2ε / (1+2ε)) / (log ∆) 2 , with (1 + ε)∆/ log(∆/ √ t) colours. This generalises and improves upon several notable results including those of Kim (1995) and Alon, Krivelevich and Sudakov (1999), and more recent ones of Molloy (2019) and Achlioptas, Iliopoulos and Sinclair (2019). This bound on the chromatic number is tight up to an asymptotic factor 2 and it coincides with a famous algorithmic barrier to colouring random graphs