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Linear amortized time enumeration algorithms for compatible Euler trails in edge-colored graphs

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Abstract

A compatible Euler trail (tour) in an edge-colored graph is an Euler trail (tour) in which each two edges traversed consecutively along the Euler trail (tour) have distinct colors. In this paper, we show that the problem of counting compatible Euler trails in edge-colored graphs is \(\#\)P-complete, and develop O(mN) time algorithms for enumerating compatible Euler trails (tours) in edge-colored graphs with m edges and N compatible Euler trails (tours). It is worth mentioning that our algorithms can run in O(N) time when there is no vertex v with degree 4 and maximum monochromatic degree 2.

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Acknowledgements

We thank the anonymous referees for their careful reading, and for comments on an earlier version that improved the presentation.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, Shaanxi, People’s Republic of China

    Yuhang Bai, Shenggui Zhang & Yandong Bai

  2. Xi’an-Budapest Joint Research Center for Combinatorics, Northwestern Polytechnical University, Xi’an, 710129, Shaanxi, People’s Republic of China

    Yuhang Bai, Shenggui Zhang & Yandong Bai

  3. School of Mathematics and Computer Science, Yan’an University, Yan’an, 716000, Shaanxi, People’s Republic of China

    Zhiwei Guo

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  1. Yuhang Bai
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  2. Zhiwei Guo
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Correspondence to Shenggui Zhang.

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Supported by NSFC (Nos. 11601430, 12071370, 12131013 and U1803263), and the Natural Science Foundation of Shaanxi Province (Nos. 2020JQ-099 and 2023-JC-QN-0004).

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Bai, Y., Guo, Z., Zhang, S. et al. Linear amortized time enumeration algorithms for compatible Euler trails in edge-colored graphs. J Comb Optim 45, 73 (2023). https://doi.org/10.1007/s10878-023-01005-w

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  • DOI: https://doi.org/10.1007/s10878-023-01005-w

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