Abstract
Let G = (V, E) be a graph with vertex set V and edge set E. Given non negative integers r, s and t, an [r, s, t]-coloring of a graph G is a proper total coloring where the neighboring elements of G (vertices and edges) receive colors with a certain difference r between colors of adjacent vertices, a difference s between colors of adjacent edges and a difference t between colors of a vertex and an incident edge. Thus [r, s, t]-colorings generalize the classical colorings of graphs and can have applications in different fields like scheduling, channel assignment problem, etc. The [r, s, t]-chromatic number χ r,s,t (G) of G is the minimum k such that G admits an [r, s, t]-coloring. In our paper we propose several bounds for the [r, s, t]-chromatic number of the cartesian and direct products of some graphs.
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References
Bazzaro F., Montassier M., Raspaud A.: (d,1)-total labelling of planar graphs with large girth and high maximum degree. Discrete Math. 307, 2141–2151 (2007)
Bottreau A., Métivier Y.: Some remarks on the Kronecker product of graphs. Inf. Process. Lett. 68, 55–61 (1998)
Chang C.Y., Clark W.E., Hare E.O.: Domination numbers of complete grid graphs I. Ars Combinatoria 36, 97–111 (1994)
Čižek N., Klavžar S.: On the chromatic number of the lexicographic product and the cartesian sum of graphs. Discrete Math. 134(1–3), 17–24 (1994)
Dekar L., Effantin B., Kheddouci H.: [r, s, t]-colorings of Trees and Bipartite graphs. Discrete Math. 310, 260–269 (2010)
Effantin B., Kheddouci H.: Grundy number of graphs. Discussiones Mathematicae Graph Theory 27(1), 5–18 (2007)
Gravier S.: Total domination number of grid graphs. Discrete Appl. Math. 121(1–3), 119–128 (2002)
Horton J.D., Wallis W.D.: Factoring the cartesian product of a cubic graph and a triangle. Discrete Math. 259(1–3), 137–146 (2002)
Jha P.K., Agnihotri N., Kumar R.: Long cycles and long paths in the Kronecker product of a cycle and a tree. Discrete Appl. Math. 74, 101–121 (1997)
Jha P.K.: Kronecker product of paths and cycles: decomposition, factorization and bi-pancyclicity. Discrete Math. 182, 153–167 (1998)
Jha P.K.: Smallest independent dominating sets in Kronecker product of cycles. Discrete Appl. Math. 113, 303–306 (2001)
Jha P.K.: Perfect r-domination in the Kronecker product of two cycles, with an application to diagonal/toroidal mesh. Inf. Process. Lett. 87, 163–168 (2003)
Kemnitz A., Marangio M.: [r, s, t]-colorings of graphs. Discrete Math. 307(2), 199–207 (2007)
Kemnitz A., Marangio M., Mihók P.: [r, s, t]-Chromatic numbers and hereditary properties of graphs. Discrete Math. 307(7–8), 916–922 (2007)
Kemnitz A., Lehmann J.: [r, s, t]-Colorings of stars. Congressus Numerantium 185, 65–80 (2007)
Kouider M., Mahéo M.: Some bound for the b-chromatic number of a graph. Discrete Math. 256, 267–277 (2002)
Ruskey, F., Sawada, J.: Bent Hamilton cycles in d-dimensional grid graphs. Electron. J. Comb. 10, #R1 (2003)
Schiermeyer I., Villà à M.S.: [r, s, t]-Colourings of paths. Opuscula Mathematica 27, 131–149 (2007)
Tigrine F., Kheddouci H.: The minimum feedback vertex set for Kronecker product of graphs. Utilitas Mathematica 74, 207–238 (2007)
Villà à, M.S.: [r, s, t]-Colourings of Paths, Cycles and Stars. Doctoral thesis, TU Bergakademie, Freiberg (2005)
Zhu X.: Star chromatic numbers and products of graphs. J. Graph Theory 16, 557–569 (1992)
Zhu X.: The fractional chromatic number of the direct product of graphs. Glasg. Math. J. 44, 103–115 (2002)
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Dekar, L., Effantin, B. & Kheddouci, H. [r, s, t]-Colorings of Graph Products. Graphs and Combinatorics 30, 1135–1147 (2014). https://doi.org/10.1007/s00373-013-1338-4
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DOI: https://doi.org/10.1007/s00373-013-1338-4