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Decomposition of complete graphs into disconnected bipartite graphs with seven edges and eight vertices | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 20، دوره 11، شماره 2، شهریور 2026، صفحه 661-684 اصل مقاله (704.85 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2024.29264.1915 | ||
| نویسندگان | ||
| Dalibor Froncek* 1؛ Michael Kubesa2 | ||
| 1Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN 55812, U.S.A | ||
| 2Department of Applied Mathematics, VŠB-Technical University of Ostrava, 17. listopadu 15, 708 33 Ostrava, Czech Republic | ||
| چکیده | ||
| In this paper, we continue investigation of decompositions of complete graphs into graphs with seven edges. The spectrum has been completely determined for such graphs with at most six vertices. Connected graphs with seven edges and seven vertices are necessarily unicyclic and the spectrum for bipartite ones was completely determined by the authors. Connected graphs with seven edges and eight vertices are trees and the spectrum was found by Huang and Rosa. As a next step in the quest of completing the spectrum for all graphs with seven edges, we completely solve the case of disconnected bipartite graphs with seven edges and eight vertices. | ||
| کلیدواژهها | ||
| graph decomposition؛ $G$-design؛ $\rho$-labeling | ||
| مراجع | ||
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[2] A. Blinco, Decompositions of complete graphs into theta graphs with fewer than ten edges, Util. Math. 64 (2003), 197–212.
[3] R.C. Bunge, On 1-rotational decompositions of complete graphs into tripartite graphs, Opuscula Math. 39 (2019), no. 5, 623–643. https://doi.org/10.7494/OpMath.2019.39.5.623
[4] S. El-Zanati and C. Vanden Eynden, On Rosa-type labelings and cyclic graph decompositions, Math. Slovaca 59 (2009), no. 1, 1–18. https://doi.org/10.2478/s12175-008-0108-x
[5] J. Fahnenstiel and D. Froncek, Decomposition of complete graphs into connected bipartite unicyclic graphs with eight edges, Electron. J. Graph Theory Appl. 7 (2019), no. 2, 235–250. https://doi.org/10.5614/ejgta.2019.7.2.4
[6] D. Froncek and M. Kubesa, Decomposition of complete graphs into connected unicyclic bipartite graphs with seven edges, Bull. Inst. Combin. Appl. 93 (2021), 52–80.
[7] C. Huang and A. Rosa, Decomposition of complete graphs into trees, Ars Combin. 5 (1978), 23–63.
[8] M. Meszka, personal communication.
[9] R.C. Read and R.J. Wilson, An Atlas of Graphs, Clarendon Press, 1998.
[10] G. Ringel, Problem 25, Theory of Graphs and its Applications, Proc. Symposium Smolenice (Prague), 1964, p. 164.
[11] A. Rosa, personal communication.
[12] A. Rosa, On certain valuations of the vertices of a graph, Theory of Graphs (Intl. Symp. Rome 1966), 1967.
[13] Z. Tian, Y. Du, and Q. Kang, Decomposing complete graphs into graphs with six vertices and seven edges, Ars Combin. 81 (2006), 257–279. | ||
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