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Characterizations for Some Types of DNA Graphs

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Abstract

Vertex induced subgraphs of directed de Bruijn graphs with labels of fixed length k and over α letter alphabet are (α,k)-labelled. DNA graphs are (4,k)-labelled graphs. Pendavingh et al. proved that it is NP-hard to determine the smallest value α k (D) for which a directed graph D can be (α k (D),k)-labelled for any fixed \(k \geqslant 3\). In this paper, we obtain the following formulas: \(\alpha_k(C_n)=\lceil\sqrt[k-1]{n}\rceil\) and\(\alpha_k(P_n)=\lceil\sqrt[k-1]{n+1}\rceil\) for cycle C n and path P n . Accordingly, we show that both cycles and paths are DNA graphs. Next we prove that rooted trees and self-adjoint digraphs admit a (Δ,k)-labelling for some positive integer k and they are DNA graphs if and only if Δ ≤ 4, where Δ is the maximum number in all out-degrees and in-degrees of such digraphs.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu, 730000, P.R. China

    Xianyue Li & Heping Zhang

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  1. Xianyue Li
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  2. Heping Zhang
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Correspondence to Heping Zhang.

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Li, X., Zhang, H. Characterizations for Some Types of DNA Graphs. J Math Chem 42, 65–79 (2007). https://doi.org/10.1007/s10910-006-9099-4

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  • DOI: https://doi.org/10.1007/s10910-006-9099-4

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