2‐Distance Coloring of Sparse Graphs. Issue 3 (27th February 2014) - Bonamy, Marthe
Abstract For graphs of bounded maximum average degree, we consider the problem of 2‐distance coloring, that is, the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbor receive different colors. We prove that graphs with maximum average degree less than 73 and maximum degree Δ at least 4 are 2‐distance (Δ+1)‐colorable, which is optimal and improves previous results from Dolama and Sopena, and from Borodin et al. We also prove that graphs with maximum average degree less than 125 (resp. 52, 187) and maximum degree Δ at least 5 (resp. 6, 8) are list 2‐distance (Δ+1)‐colorable, which improves previous results from Borodin et al., and from Ivanova. We prove that any graph with maximum average degree m less than 145 and with large enough maximum degree Δ (depending only on m) can be list 2‐distance (Δ+1)‐colored. There exist graphs with arbitrarily large maximum degree and maximum average degree less than 3 that cannot be 2‐distance (Δ+1)‐colored: the question of what happens between 145 and 3 remains open. We prove also that any graph with maximum average degree m<4 can be list 2‐distance (Δ+C)‐colored (C depending only on m). It is optimal as there exist graphs with arbitrarily large maximum degree and maximum average degree less than 4 that cannot be 2‐distance colored with less than 3Δ2 colors. Most of the above results can be transposed to injective list coloring with one color less.