Good and semi-strong colorings of oriented planar graphs
Information Processing Letters
Randomized algorithms
Acyclic and oriented chromatic numbers of graphs
Journal of Graph Theory
The chromatic number of oriented graphs
Journal of Graph Theory
On the oriented chromatic number of grids
Information Processing Letters
Tree-depth, subgraph coloring and homomorphism bounds
European Journal of Combinatorics
A constructive proof of the general lovász local lemma
Journal of the ACM (JACM)
Random Structures & Algorithms
Bounds on vertex colorings with restrictions on the union of color classes
Journal of Graph Theory
Bounds on Edge Colorings with Restrictions on the Union of Color Classes
SIAM Journal on Discrete Mathematics
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We present an improved upper bound of O(d^1^+^1^m^-^1) for the (2,F)-subgraph chromatic number @g"2","F(G) of any graph G of maximum degree d. Here, m denotes the minimum number of edges in any member of F. This bound is tight up to a (logd)^1^/^(^m^-^1^) multiplicative factor and improves the previous bound presented in Aravind and Subramanian (2011) [5]. Along the way, we state and prove an easy-to-use version of the Lovasz Local Lemma. We also obtain a relationship connecting the oriented chromatic number @g"o(G) of graphs and the (j,F)-subgraph chromatic numbers @g"j","F(G) introduced and studied in Aravind and Subramanian (2011) [5]. In particular, we relate the oriented chromatic number and the (2,r)-treewidth chromatic number and show that @g"o(G)@?k((r+1)2^r)^k^-^1 for any graph G having (2,r)-treewidth chromatic number at most k. The latter parameter is the least number of colors in any proper vertex coloring which is such that the subgraph induced by the union of any two color classes has treewidth at most r. We also generalize a result of Alon et al. (1996) [3] on acyclic chromatic number of graphs on surfaces (with characteristic -@c@?0) to (2,F)-subgraph chromatic numbers and prove that @g"2","F(G)=O(@c^m^/^(^2^m^-^1^)) for some constant m depending only on F. We also show that this bound is nearly tight. We then use this result to show that graphs of genus g have oriented chromatic number at most 2^O^(^g^^^1^^^/^^^2^^^+^^^@e^) for every fixed @e0. We also refine the proof of a bound on @g"o(G) obtained by Kostochka et al. (1997) in [10] to obtain an improved bound on @g"o(G).