| Graph labelling traces its origin to the famous conjecture that all trees are graceful presentedby A.Rosa in 1966. Vertex labelling is a mapping that maps the vertex set into integer set. Edge labelling is a mapping that maps the edge set into integer set. According to the different requirements for the mapping, many variations of graph labellings have been evolved. In this paper, two classes of graph labellings: (d, 1)-total labelling, (2, 1)-labelling are researched.Yeh and then Griggs first consider (2, 1)-labelling. (2, 1)-labelling is motivated by the radio channels assignment problem in computer network. Using nonnegative integers to representchannels, so that close locations receive different channels, and channels for very close locations are at least two apart such that these channels would not interfere with each other.(d, 1)-total labelling is motivated from (2, 1)-labelling.Havet provedλdT(G)≥d+r for r-regular graph. In this paper, we prove thatλdT(G)≥d+r+1 for r-regular nonbipartite graph with d≥r≥3.(d,1)-total numbers of the generalized Petersen graphs, Flower Snark graphs and its relatedgraphs and Goldberg Snark graphs and its related graphs are researched in this paper, and we obtain the following results:(1) For even n and odd k,λd≥2T(P(n,k))=d+3.(2) For odd n or even k,λd≥3T(P(n,k))=d+4.(3) For even n,λd≥2T(Hn)=λd≥2T(Gn)=d+3.(4)For odd n,λ2T(Hn)=λ2T(Gn)=5,λd≥3T(Hn)=λd≥3T(Gn)=d+4.(5)λ2T(Gk)=λ2T(TGk)=5,λd≥3T(Gk)=λd≥3T(TGk)=d+4.(2, 1)-labelling numbers of Flower Snark graphs and its related graphs are researched in this paper, and we obtain the following conclusions:(1)For n>3,λ(Hn) =λ(Gn) = 6.(2)For n= 3,λ(Hn) = 7,λ(Gn) = 6. |
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