Journal of Formalized Mathematics
Volume 9, 1997
University of Bialystok
Copyright (c) 1997 Association of Mizar Users

Euler Circuits and Paths


Yatsuka Nakamura
Shinshu University, Nagano
Piotr Rudnicki
University of Alberta, Edmonton

Summary.

We prove the Euler theorem on existence of Euler circuits and paths in multigraphs.

This work was partially supported by NSERC Grant OGP9207 and Shinshu Endowment Fund for Information Science.

MML Identifier: GRAPH_3

The terminology and notation used in this paper have been introduced in the following articles [16] [7] [20] [17] [21] [4] [6] [5] [8] [10] [2] [18] [12] [3] [9] [19] [1] [13] [14] [11] [15]

Contents (PDF format)

  1. Preliminaries
  2. Degree of a vertex
  3. Adding an edge to a graph
  4. Some properties of and operations on cycles
  5. Euler circuits and paths

Acknowledgments

We would like to thank Cz. Byli\'nski for many helpful comments.

Bibliography

[1] Grzegorz Bancerek. Cardinal numbers. Journal of Formalized Mathematics, 1, 1989.
[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Journal of Formalized Mathematics, 1, 1989.
[3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Journal of Formalized Mathematics, 1, 1989.
[4] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[5] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[6] Czeslaw Bylinski. Partial functions. Journal of Formalized Mathematics, 1, 1989.
[7] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[8] Czeslaw Bylinski. A classical first order language. Journal of Formalized Mathematics, 2, 1990.
[9] Czeslaw Bylinski. Finite sequences and tuples of elements of a non-empty sets. Journal of Formalized Mathematics, 2, 1990.
[10] Czeslaw Bylinski. The modification of a function by a function and the iteration of the composition of a function. Journal of Formalized Mathematics, 2, 1990.
[11] Czeslaw Bylinski and Piotr Rudnicki. The correspondence between monotonic many sorted signatures and well-founded graphs. Part I. Journal of Formalized Mathematics, 8, 1996.
[12] Agata Darmochwal. Finite sets. Journal of Formalized Mathematics, 1, 1989.
[13] Krzysztof Hryniewiecki. Graphs. Journal of Formalized Mathematics, 2, 1990.
[14] Yatsuka Nakamura and Piotr Rudnicki. Vertex sequences induced by chains. Journal of Formalized Mathematics, 7, 1995.
[15] Piotr Rudnicki and Andrzej Trybulec. Abian's fixed point theorem. Journal of Formalized Mathematics, 9, 1997.
[16] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[17] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.
[18] Michal J. Trybulec. Integers. Journal of Formalized Mathematics, 2, 1990.
[19] Wojciech A. Trybulec. Pigeon hole principle. Journal of Formalized Mathematics, 2, 1990.
[20] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[21] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.

Received July 29, 1997

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