Coloring three graph classes with Δ − 1 colors

The Borodin–Kostochka conjecture states that every connected graph [Formula: see text] with [Formula: see text] satisfies [Formula: see text]. In this paper, we verify this conjecture for any [Formula: see text]-free graph [Formula: see text] (where a [Formula: see text] is a [Formula: see text] with a pendant edge), by further prohibiting induced subgraph HVN (where a [Formula: see text] is a [Formula: see text] together with one more vertex which is adjacent to exactly two vertices of [Formula: see text]), we verify this conjecture with [Formula: see text]. Furthermore, for any [Formula: see text]-free graph [Formula: see text] (where a [Formula: see text] is a [Formula: see text] with a pendant edge), we verify this conjecture with [Formula: see text].