Abstract
We prove that the edges of every even graph G = G 1 + G 2 that is the join of two regular graphs G i = (V i ,E i ) can be coloured with Δ(G) colours, whenever Δ(G) = Δ(G 2) + |V(G 1)|. The proof of this result yields a combinatorial algorithm to optimally colour the edges of this type of graphs.
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References
Berry, R., Modiano, E.: Optimal Transceiver Scheduling in WDM/TDM Networks. IEEE J. on Select. Areas in Comm. 23, 1479–1495 (2005)
Chvátal, V.: On certain polytopes associated with graphs. J. Combin. Theory, Ser. B 18, 138–154 (1975)
Chetwynd, A.G., Hilton, A.J.W.: Star multigraphs with three vertices of maximum degree. Math. Proc. Cambridge Phil. Soc. 100, 303–317 (1986)
Chetwynd, A.G., Hilton, A.J.W.: The edge-chromatic class of graphs with maximum degree at least |V| − 3. Annals of Discrete Mathematics 41, 91–110 (1989)
Corneil, D.G., Lerchs, H., Burlingham, L.S.: Complement reducible graphs. Discrete Appl. Math. 3, 163–174 (1981)
Corneil, D.G., Perl, Y., Stewart, L.K.: A linear recognition algorithm for cographs. SIAM J. of Comput. 14, 926–934 (1985)
De Werra, D.: Investigations on an edge-coloring problem. Discrete Math. 1, 167–179 (1972)
De Simone, C., Galluccio, A.: Edge-colouring of regular graphs of large degree. Theor. Comp. Sc. 389, 91–99 (2007)
De Simone, C., de Mello, C.P.: Edge colouring of join graphs. Theor. Comp. Sc. 355, 364–370 (2006)
Holyer, I.: The NP-completeness of edge-colouring. SIAM J. Comput. 14, 718–720 (1981)
Hoffman, D.G., Rodger, C.A.: The chromatic index of complete multipartite graphs. J. Graph Theory 16, 159–163 (1992)
Möhring, R.H.: Algorithmic aspects of the substitution decomposition in optimization over relations, set systems and Boolean functions. Ann. Oper. Res. 4, 195–225 (1985)
McDiarmid, C.J.H.: The solution of a timetabling problem. J. Inst. Math. Appl. 9, 23–34 (1972)
Perkovic, L., Reed, B.: Edge coloring regular graphs of high degree. Discrete Math. 165/166, 567–570 (1997)
Perkovic, L.: Edge Coloring, Polyhedra and Probability, Ph.D Thesis Carnegie Mellon Univ., US (1998)
Vizing, V.G.: On an estimate of the chromatic class of a p-graph. Diskret Analiz 3, 25–30 (1964) (in Russian)
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De Simone, C., Galluccio, A. (2008). A Combinatorial Algorithm to Optimally Colour the Edges of the Graphs That Are Join of Regular Graphs. In: Yang, B., Du, DZ., Wang, C.A. (eds) Combinatorial Optimization and Applications. COCOA 2008. Lecture Notes in Computer Science, vol 5165. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85097-7_33
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DOI: https://doi.org/10.1007/978-3-540-85097-7_33
Publisher Name: Springer, Berlin, Heidelberg
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