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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2024, Vol. 41 ›› Issue (2): 311-325.doi: 10.3969/j.issn.1005-3085.2024.02.008

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探索对奇边优美差全着色封闭的图格

张明军1,2,  杨见青1,2,  姚  兵3   

  1. 1. 兰州财经大学信息工程与人工智能学院，兰州 730020;
    2. 甘肃省电子商务技术与应用重点实验室，兰州 730020;
    3. 西北师范大学数学与统计学院，兰州 730070
  • 收稿日期:2022-07-14 接受日期:2023-07-31 出版日期:2024-04-15 发布日期:2024-06-15
  • 基金资助:
    国家自然科学基金 (61662066)；兰州财经大学高等教育研究项目 (LJZ202309)；兰州财经大学科研资助项目 ,(Lzufe2022B-002)；兰州财经大学中国西北金融研究中心项目 (JYYZ201905).

Graphic Lattices Having the Closeness of $W$-type Colorings

ZHANG Mingjun1,2,  YANG Jianqing1,2,  YAO Bing3   

  1. 1. School of Information Engineering and Artificial Intelligence, Lanzhou University of Finance and Economics, Lanzhou 730020;
    2. Key Laboratory of E-Business Technology and Application of Gansu Province, Lanzhou 730020;
    3. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070
  • Received:2022-07-14 Accepted:2023-07-31 Online:2024-04-15 Published:2024-06-15
  • Supported by:
    The National Natural Science Foundation of China (61662066); the Higher Education Research Project of Lanzhou University of Finance and Economics (LJZ202309); the Research Funding Project of Lanzhou University of Finance and Economics (Lzufe2022B-002); the Northwest China Financial Research Center Project of Lanzhou University of Finance and Economics (JYYZ201905).

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摘要：

为深入拓扑编码的研究，定义了新的图全标号和图全着色：(集有序) 奇边优美差全标号/全着色，孪生 (集有序) 奇边优美差全标号/全着色。证明了若偶图 $T$ 承认集有序奇优美标号，则给偶图 $T$ 添加 $m$ 片叶子后得到的偶图 $T^*$ 承认一个奇边优美差全着色；每棵树承认一个奇边优美差全着色。定理的证明均可转化为可行、有效的算法。为建立随机着色的图格，给出随机添加叶子的奇边优美差全着色算法和一致-$k^*$ 优美差算法，建立了对奇边优美差全着色封闭的一致-$k^*$ 优美差图格、孪生一致-$(k^*,n^*)$ 优美差图格，以及一个图格同态到另一个图格的图格同态。

关键词: 格密码, 拓扑编码, 奇边优美差全着色, 图格, 非对称密码学

Abstract:

For deeply investigating topological coding, we define new graph total labelings/total colorings: (set-ordered) odd-edge graceful-difference total labelings/total colorings, twin (set-ordered) odd-edge graceful-difference total labelings/total colorings. We prove two results as follows: If bipartite graph $T$ admits a set-ordered odd-graceful labeling, then the bipartite graph $T^*$ obtained by adding $m$ leaves to $T$ admits an odd-edge graceful-difference total coloring; Each tree admits an odd-edge graceful-difference total coloring. For building randomly graph lattices, we present the algorithm of odd-edge graceful-difference total coloring based on adding randomly leaves and the uniformly $k^*$ graceful-difference algorithm, and make uniformly $k^*$ graceful-difference graph lattices, twin uniformly $(k^*,n^*)$ graceful-difference graph lattices, as well as a graphic lattice is homomorphism to another graphic lattice, called graphic-lattice homomorphism.

Key words: lattice-based cryptography, topological coding, odd-edge graceful-difference total coloring, graphic lattice, asymmetric cryptography

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