2023/04/18 by Alexander Clow, Clow, Alexander, Ladislav Stacho +1 · 1 citation
Computer Science · Engineering · #Graph Labeling and Dimension Problems #Advanced Graph Theory Research #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2304.09320
This paper considers upper bounds on the oriented chromatic number χo(G), of an oriented graph G in terms of its 2-dipath chromatic number χ2(G), degeneracy d(G), and maximum degree Δ(G). In particular, we show that for all graphs G with χ2(G) ≤ k where k ≥ 2 and d(G) ≤ t where t ≥ log2(k), χo(G) = 33/10(k t2 2t). This improves an upper bound of MacGillivray, Raspaud, and Swartz of the form χo(G) ≤ 2χ2(G) -1 to a polynomial upper bound for many classes of graphs, in particular, those with bounded degeneracy. Additionally, we asymptotically improve bounds for the oriented chromatic number in terms of maximum degree and degeneracy. For instance, we show that χo(G) ≤ (2ln2 +o(1))Δ2 2Δ for all graphs, and χo(G) ≤ (2+o(1))Δd 2d for graphs where degeneracy grows sublinearly in maximum degree. Here the asypmtotics are in Δ. The former improves the asymptotics of a results by Kostochka, Sopena, and Zhu \citekostochka1997acyclic, while the latter improves the asymptotics of a result by Aravind and Subramanian \citearavind2009forbidden. Both improvements are by a constant factor.