Abstract
In this study, rather than transitioning directly from geometric structures to graph theory, we have derived several general results and theorems concerning the coloring of points and lines within affine and projective structures. We approached this topic through the lens of vertex and edge coloring concepts, pivotal subjects within graph theory. Our investigation sheds light on the intricate relationship between geometric structures and graph theory, providing a novel perspective on coloring methodologies. Extending the principles of vertex and edge coloring to affine and projective spaces, we uncover fundamental insights into the interplay between geometry and combinatorial mathematics.
Keywords
References
- Erdös P., On The Combinatorial Problems Which I Would Most Like To See Solved, Combinatorica pages 25–42, https://doi.org/10.1007/BF02579174, (1981).
- Beutelspacher, A., Jungnickel, D.; Vanstone, S.A., On the chromatic index of a finite projective space, Geom Dedicata 32, 313–318, https://doi.org/10.1007/BF00147923, (1989).
- Araujo-Pardo, G., Kiss, G., Rubio-Montiel, C., Vázquez-Avila, A., On chromatic indices of finite affine spaces, arXiv preprint, arXiv:1711.09031, (2017).
- Xu, L., Feng, T., The chromatic index of finite projective spaces, J. Combin. Des., 31, 432–446, https://doi.org/10.1002/jcd.21904, (2023).
- Meszka, M., The Chromatic Index of Projective Triple Systems, J. Combin. Designs, 21: 531-540, https://doi.org/10.1002/jcd.21368, (2013).
- Ozeki, K., Kempe Equivalence Classes of Cubic Graphs Embedded on the Projective Plane, Combinatorica, 42 (Suppl 2), 1451–1480, https://doi.org/10.1007/s00493-021-4330-2, (2022).
- Hall, M., Projective Planes, Trans. Am. Math. Soc., 54, 229-77, (1943) and correction, 65, 473-4, (1949).
- Batten, L. M., Combinatorics of Finite Geometries, 2nd edition, Cambridge University Press: New York, (1997).
- Pickert, G., Projektive Ebenen, Springer-Verlag: Berlin, Gottingen, Heidelberg, (1955).
- Hughes, D. R., Piper, F. C., Projective Planes, Springer: New York, (1973).
- Bennett, M. K., Affine and Projective Geometry, John Wiley-Interscience: New York, (1995).
- Faulkner, T. E., Projective Geometry, Dover: New York, (1949).
- Lam, C. W. H., Thiel, L., Swiercz, S., The Non-Existence of Finite Projective Planes of Order 10. Canadian Journal of Mathematics, 41-6: 1117–1123, https://doi.org/10.4153/CJM-1989-049-4, (1989).
- Kaya, R., Projektif Geometri, Osmangazi Üni. Yayınları: Eskişehir, (2005).
- Jungnickel, D., Graphs, Networks and Algorithms, Springer: Augsburg, (2013).
- Bondy, A., Murty, U.S.R., Graph Theory, Springer: London, (2008).
Abstract
Bu çalışmada, doğrudan geometrik yapılardan graf teorisine geçiş yapmak yerine, afin ve projektif yapılardaki nokta ve doğruların renklendirilmesiyle ilgili birkaç genel sonuç ve teorem ürettik. Bu konuya, graf teorideki temel konular olan köşe ve kenar renklendirme kavramları merceğinden yaklaştık. Araştırmamız, geometrik yapılar ve graf teori arasındaki karmaşık ilişkiye ışık tutarak, renklendirme metodolojilerine yeni bir bakış açısı sağlıyor. Köşe ve kenar renklendirme ilkelerini afin ve projektif uzaylara genişleterek, geometri ve kombinatoryal matematik arasındaki etkileşime dair temel içgörüler ortaya çıkarıyoruz.
Keywords
References
- Erdös P., On The Combinatorial Problems Which I Would Most Like To See Solved, Combinatorica pages 25–42, https://doi.org/10.1007/BF02579174, (1981).
- Beutelspacher, A., Jungnickel, D.; Vanstone, S.A., On the chromatic index of a finite projective space, Geom Dedicata 32, 313–318, https://doi.org/10.1007/BF00147923, (1989).
- Araujo-Pardo, G., Kiss, G., Rubio-Montiel, C., Vázquez-Avila, A., On chromatic indices of finite affine spaces, arXiv preprint, arXiv:1711.09031, (2017).
- Xu, L., Feng, T., The chromatic index of finite projective spaces, J. Combin. Des., 31, 432–446, https://doi.org/10.1002/jcd.21904, (2023).
- Meszka, M., The Chromatic Index of Projective Triple Systems, J. Combin. Designs, 21: 531-540, https://doi.org/10.1002/jcd.21368, (2013).
- Ozeki, K., Kempe Equivalence Classes of Cubic Graphs Embedded on the Projective Plane, Combinatorica, 42 (Suppl 2), 1451–1480, https://doi.org/10.1007/s00493-021-4330-2, (2022).
- Hall, M., Projective Planes, Trans. Am. Math. Soc., 54, 229-77, (1943) and correction, 65, 473-4, (1949).
- Batten, L. M., Combinatorics of Finite Geometries, 2nd edition, Cambridge University Press: New York, (1997).
- Pickert, G., Projektive Ebenen, Springer-Verlag: Berlin, Gottingen, Heidelberg, (1955).
- Hughes, D. R., Piper, F. C., Projective Planes, Springer: New York, (1973).
- Bennett, M. K., Affine and Projective Geometry, John Wiley-Interscience: New York, (1995).
- Faulkner, T. E., Projective Geometry, Dover: New York, (1949).
- Lam, C. W. H., Thiel, L., Swiercz, S., The Non-Existence of Finite Projective Planes of Order 10. Canadian Journal of Mathematics, 41-6: 1117–1123, https://doi.org/10.4153/CJM-1989-049-4, (1989).
- Kaya, R., Projektif Geometri, Osmangazi Üni. Yayınları: Eskişehir, (2005).
- Jungnickel, D., Graphs, Networks and Algorithms, Springer: Augsburg, (2013).
- Bondy, A., Murty, U.S.R., Graph Theory, Springer: London, (2008).
Details
| Primary Language | English |
|---|---|
| Subjects | Combinatorics and Discrete Mathematics (Excl. Physical Combinatorics), Pure Mathematics (Other) |
| Journal Section | Research Article |
| Authors | |
| Submission Date | November 11, 2024 |
| Acceptance Date | May 5, 2025 |
| Early Pub Date | July 11, 2025 |
| Publication Date | July 15, 2025 |
| Published in Issue | Year 2025 Volume: 27 Issue: 2 |