Hard-to-color Graphs for Connected Sequential Colorings

Abstract The sequential coloring method colors the vertices of a graph in a given order assigning each vertex the smallest available color. A sequential coloring is called connected-coloring if at any time the colored vertices induce a connected graph. A graph G is said to be hard-to-color if every connected-coloring produces a nonoptimal coloring and partially hard-to-color if every such coloring starting in a specified vertex v is nonoptimal. We present smallest partially hard- to-color graphs. Further a hard-to-color graph with 18 vertices is stated which is believed to be the smallest graph with this property. We prove that it is the smallest cubic hard-to-color graph.