Broadcast Chromatic Numbers of Graphs 1 ... - ResearchGate

Proposition 7.1 If χ b (G) ≥ diam(G)+x, then χ b (G✷K 2 ) ≥ diam(G✷K 2 )+2x − 1.Proof.The result is true for x ≤ 0, since χ b (G✷K 2 ) ≥ χ b (G). So assume x ≥ 1.Suppose there exists a broadcast coloring π of G✷K 2 that uses at mostdiam(G✷K 2 ) + 2x − 2 colors. It follows that the 2x − 1 biggest colors—callthem Z—are used at most once in G✷K 2 . It follows that one of the copiesof G is broadcast-colored by the colors up to diam(G✷K 2 ) − 1 = diam(G)together with at most x − 1 colors of Z. Thus χ b (G) ≤ diam(G) + x − 1, acontradiction. ✷For example, this shows that the broadcast chromatic number of thecube is at least a positive fraction of its order. Next is a result regardingthe first few hypercubes Q d .Proposition 7.2 χ b (Q 1 ) = 2, χ b (Q 2 ) = 3, χ b (Q 3 ) = 5, χ b (Q 4 ) = 7, andχ b (Q 5 ) = 15.Proof. The values for Q 1 and Q 2 follow from earlier results. ConsiderQ 3 . The upper bound is from Proposition 2.1. To see that five colors arerequired, note that since β 0 (Q 3 ) = 4, at most four vertices can be colored1. Further, at most two vertices can be colored 2; but, if four vertices arecolored 1 then no more than one vertex can be colored 2. Therefore, thenumber of vertices colored 1 or 2 must be at most five. Since diam(Q 3 ) = 3,no color greater than 2 can be used more than once. As there are eightvertices in Q 3 , this means that at least five colors are required. The lowerbound for Q 4 follows from the value for Q 3 by the above proposition.For a suitable broadcast coloring in each case, use the greedy algorithmas follows. Place color 1 on a maximum independent set; then color withcolor 2 as many as possible, then color 3 and so on. ✷We look next at the asymptotics. Bounds for the packing numbers of thehypercubes are well explored in coding theory. For our purposes it suffices18