Coloring some (P₆, C₄) -free graphs with -1 colors

The Borodin-Kostochka Conjecture states that for a graph G, if (G) 9, then (G) \ (G) -1, (G) \. We use Pₜ and Cₜ to denote a path and a cycle on t vertices, respectively. Let C=v₁v₂v₃v₄v₅v₁ be an induced C₅. A C₅^+ is a graph obtained from C by adding a C₃=xyzx and a P₂=t₁t₂ such that (1) x and y are both exactly adjacent to v₁, v₂, v₃ in V (C), z is exactly adjacent to v₂ in V (C), t₁ is exactly adjacent to v₄, v₅ in V (C) and t₂ is exactly adjacent to v₁, v₄, v₅ in V (C), (2) t₁ is exactly adjacent to z in \x, y, z\ and t₂ has no neighbors in \x, y, z\. In this paper, we show that the Borodin-Kostochka Conjecture holds for (P₆, C₄, H) -free graphs, where H \K₇, C₅^+\. This generalizes some results of Gupta and Pradhan in GP21, GP24.