Geodesic
Hamiltonian completion
Trudy Instituta matematiki, Tome 19 (2011) no. 2, pp. 87-90 
O. V. Maksimovich; R. I. Tyshkevich. Hamiltonian completion. Trudy Instituta matematiki, Tome 19 (2011) no. 2, pp. 87-90. http://geodesic.mathdoc.fr/item/TIMB_2011_19_2_a9/
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Voir la notice de l'article provenant de la source Math-Net.Ru

This article is a continuation of the work started in [1], where $L(2,1)$-coloring problem is interpreted as optimization task on the set of graph vertices. This approach enabled us to reduce solution of hamiltonian cycle problem to injective $\lambda$-coloring. Here we calculate edge distance from the given graph to the nearest graph containing hamiltonian path, also we construct hamiltonian graphs with at most one extra edge.

[1] Maksimovich O.V., Tyshkevich R.I., “In'ektivnaya $L(2,1)$-raskraska kak optimizatsionnaya zadacha na mnozhestve perestanovok vershin grafa: dominantno-porogovye grafy”, Tr. In-ta matematiki NAN Belarusi, 17:1 (2009), 110–118 | MR | Zbl

[2] Emelichev V.A., Melnikov O.I., Sarvanov V.I., Tyshkevich R.I., Lektsii po teorii grafov, 2-e izd., Nauka, M., 1990. | MR | Zbl

[3] Griggs J.R., Yeh R.K., “Labelling graphs with a condition at distance 2”, SIAM Journal of Discreet Mathematics, 5(4) (1992), 586–595 | DOI | MR | Zbl

[4] Yeh R.K., Labelling graphs with a condition at distance two, Ph.D. Thesis, Department of Mathematics, University of South Carolina, 1990

[5] Bodlaender H.L., Kloks T., Tan R.B., van Leeuwen J., “Approximations for $\lambda$-Coloring of Graphs”, The Computer Journal, 47(2) (2004), 193–204 | DOI | Zbl

[6] Bodlaender H.L., Kloks T., Tan R.B., van Leeuwen J., $\lambda$-coloring of graphs, Tech Report. Dept. of Computer Science, Utrecht University, 1999

[7] Fiala J., Kloks T., Kratochvil J., “Fixed-Parameter Complexity of $\lambda$-Labelings”, Lecture Notes in Computer Science, 1665, 1999, 350–363 | MR | Zbl