aa r X i v : . [ m a t h . G R ] N ov A NOTE ON HAYNES-HEDETNIEMI-SLATER CONJECTURE
TOMOO YOKOYAMA
Abstract.
We notice that Haynes-Hedetniemi-Slater Conjecture is true (i.e. γ ( G ) ≤ δ δ − n for every graph G of size n with minimum degree δ ≥
4, where γ ( G ) is the domination number of G ). Background and Remarks
The domination number γ ( G ) of a (finite, undirected and simple) graph G =( V, E ) is the minimum cardinality of a set D ⊆ V of vertices such that every vertexin V − D has a neighbour in D .Ore [O62] proved that γ ( G ) ≤ n for every graph G of size n with minimumdegree δ ≥
1. Blank [B73] proved that γ ( G ) ≤ n for and all but 7 exceptionalgraphs G of size n with δ ≥
2. Blank’s result was also discovered by McCuaig andShepherd [MS89]. Reed [R96] proved that γ ( G ) ≤ n for every graph G of size n with δ ≥
3. All these bounds are best possible.T. W. Haynes, S. T. Hedetniemi and P. J. Slater [HHS98] conjectured as follows:For a graph G of size n with minimal degree δ ≥ γ ( G ) ≤ δ δ − n .However, Caro and Roditty proved [CR85] [CR90] that for any graph G of size n with minimum degree δ , γ ( G ) ≤ n (1 − δ ( δ +1 ) δ ). For δ ≥
7, it is easy toverify that 1 − δ ( δ +1 ) δ < δδ − , by using calculus. Recently, the conjecture wasconfirmed for k = 4 by M. Y. Sohn and X. Yuan [SY09] and for k = 5 by H. M.Xing, L. Sun and X. G. Chen [XSC06]. Moreover, H.M. Xing, L. Sun, X. G. Chenprove that if G is a Hamiltonian graph of order n with δ ≥
6, then the conjecturewas confirmed for G . Therefore, the conjecture was open for graphs with δ = 6.However, W. E. Clark, B. Shekhtman, and S. Suen, proved [CSS98] that for anygraph G of size n with δ , γ ( G ) ≤ n (1 − Q δ +1 i =1 iδiδ +1 ). For δ = 6, it is easy to verifythat 1 − Q δ +1 i =1 iδiδ +1 < δ δ − , by using calculus. Indeed,1 − δ +1 Y i =1 iδiδ + 1 < . <
617 = δ δ − Date : January 13, 2018.
Key words and phrases. domination number, minimal degree. and 1 − δ +1 Y i =1 iδiδ + 1 − δ δ − − Y i =1 i i + 1= − · · · · · · · − · · · · · · · · · − − . ...< δ <
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