Broadcast Chromatic Numbers of Graphs 1 ... - ResearchGate

ming these appropriately, trimming leaves that are in V 1 , and then seeingwhether what remains with the partial coloring is a subdivision of a bipartitegraph. We omit the details.4 Intractable ColoringsIn contrast to the above, the problem of determining whether a graph hasa broadcast 4-coloring is intractable. We will need the following generalization.For a sequence of positive integers s 1 ≤ s 2 ≤ . . . ≤ s k , an(s 1 , s 2 , . . .,s k )-coloring is a weak partition π = (V 1 , V 2 , . . .,V k ), where V jis an s j -packing for 1 ≤ j ≤ k. Then we define the decision problem:(s 1 , s 2 , . . .,s k )-ColoringInstance: Graph GQuestion: Does G have an (s 1 , s 2 , . . .,s k )-coloring?For example, a 3-coloring is a (1, 1, 1)-coloring. The Broadcast 4-Coloring problem is equivalent to the (1, 2, 3, 4)-Coloring problem. Wewill need the intractability of a special 3-coloring problem.Proposition 4.1 (1, 1, 2)-Coloring is NP-hard.Proof. The proof is by reduction from normal 3-coloring. The reductionis to form G ′ from G as follows. Replace each edge uv by the following:add a pentagon P uv and join u, v to nonadjacent vertices of P uv ; add apentagon Q uv and join u, v to adjacent vertices of Q uv and add an edgejoining two degree-two vertices of Q uv . (See Figure 1.) The vertices u, v arecalled original in G ′ .We claim that G ′ has a (1, 1, 2)-coloring iff G is 3-colorable.Assume that G has a 3-coloring with colors red, blue and gold. We will3-color G ′ such that the gold vertices form a 2-packing. Start by giving theoriginal vertices of G ′ their color in G. For each two adjacent vertices u, vof G, color gold one vertex from each of P uv and Q uv chosen as follows.8