Pushable chromatic number of graphs with degree constraints
Pushable homomorphisms and the pushable chromatic number χp of oriented graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably observed that, for any oriented graph G⃗, we have χp(G⃗)≤χo(G⃗)≤2χp(G⃗), where χo(G⃗) denotes the oriented chromatic number of G⃗. This stands as the first general bounds on χp. This parameter was further studied in later works. This work is dedicated to the pushable chromatic number of oriented graphs fulfilling particular degree conditions. For all Δ≥29, we first prove that the maximum value of the pushable chromatic number of a connected oriented graph with maximum degree Δ lies between [Formula presented] and (Δ−3)⋅(Δ−1)⋅2Δ−1+2 which implies an improved bound on the oriented chromatic number of the same family of graphs. For subcubic oriented graphs, that is, when Δ≤3, we then prove that the maximum value of the pushable chromatic number is 6 or 7. We also prove that the maximum value of the pushable chromatic number of oriented graphs with maximum average degree less than 3 lies between 5 and 6. The former upper bound of 7 also holds as an upper bound on the pushable chromatic number of planar oriented graphs with girth at least 6.
Bensmail, Julien; Das, Sandip; Nandi, Soumen; Paul, Soumyajit; Pierron, Théo; Sen, Sagnik; and Sopena, Éric, "Pushable chromatic number of graphs with degree constraints" (2021). Journal Articles. 2251.
Open Access, Green