Spanning tree congestion of rook's graphs

Kyohei Kozawa ; Yota Otachi

Discussiones Mathematicae Graph Theory, Tome 31 (2011), p. 753-761 / Harvested from The Polish Digital Mathematics Library

Résumé

Let G be a connected graph and T be a spanning tree of G. For e ∈ E(T), the congestion of e is the number of edges in G joining the two components of T - e. The congestion of T is the maximum congestion over all edges in T. The spanning tree congestion of G is the minimum congestion over all its spanning trees. In this paper, we determine the spanning tree congestion of the rook's graph Kₘ ☐ Kₙ for any m and n.

Détail

Publié le : 2011-01-01

Zbl 1255.05047

EUDML-ID : urn:eudml:doc:270973

Bibliographie

[000] [1] J. Balogh and D. Mubayi and A. Pluhár, On the edge-bandwidth of graph products, Theoret. Comput. Sci. 359 (2006) 43-57, doi: 10.1016/j.tcs.2006.01.046. | Zbl 1098.05070

[001] [2] A. Bekmetjev and G. Hurlbert, The pebbling threshold of the square of cliques, Discrete Math. 308 (2008) 4306-4314, doi: 10.1016/j.disc.2007.08.012. | Zbl 1152.05050

[002] [3] S.L. Bezrukov, Edge isoperimetric problems on graphs, in: L. Lovasz, A. Gyarfas, G.O.H. Katona, A. Recski and L. Szekely, eds, Graph Theory and Combinatorial Biology, 7 Bolyai Soc. Math. Stud. 157-197 Janos Bolyai Math. Soc. (Budapest, 1999). | Zbl 0927.05080

[003] [4] H.L. Bodlaender, K. Kozawa, T. Matsushima and Y. Otachi, Spanning Tree Congestion of k-outerplanar Graphs, in: WAAC 2010 (2010) 34-39. | Zbl 1223.05017

[004] [5] A. Castejón and M.I. Ostrovskii, Minimum congestion spanning trees of grids and discrete toruses, Discuss. Math. Graph Theory 29 (2009) 511-519, doi: 10.7151/dmgt.1461. | Zbl 1193.05058

[005] [6] M. Chudnovsky and N. Robertson and P. Seymour and R. Thomas, The strong perfect graph theorem, Ann. of Math. 164 (2006) 51-229, doi: 10.4007/annals.2006.164.51. | Zbl 1112.05042

[006] [7] A.J. Hoffman, On the Line Graph of the Complete Bipartite Graph, Ann. Math. Statist. 35 (1964) 883-885, doi: 10.1214/aoms/1177703593. | Zbl 0127.39302

[007] [8] S.W. Hruska, On tree congestion of graphs, Discrete Math. 308 (2008) 1801-1809, doi: 10.1016/j.disc.2007.04.030. | Zbl 1151.05014

[008] [9] K. Kozawa and Y. Otachi and K. Yamazaki, On spanning tree congestion of graphs, Discrete Math. 309 (2009) 4215-4224, doi: 10.1016/j.disc.2008.12.021. | Zbl 1232.05116

[009] [10] C. Löwenstein and D. Rautenbach and F. Regen, On spanning tree congestion, Discrete Math. 309 (2009) 4653-4655, doi: 10.1016/j.disc.2009.01.012. | Zbl 1213.05093

[010] [11] R. Laskar and C. Wallis, Chessboard graphs, related designs, and domination parameters, J. Statist. Plann. Inference 76 (1999) 285-294, doi: 10.1016/S0378-3758(98)00132-3. | Zbl 0939.05065

[011] [12] H.-F. Law, Spanning tree congestion of the hypercube, Discrete Math. 309 (2009) 6644-6648, doi: 10.1016/j.disc.2009.07.007. | Zbl 1179.05032

[012] [13] J.H. Lindsey II, Assignment of numbers to vertices, Amer. Math. Monthly 71 (1964) 508-516, doi: 10.2307/2312587.

[013] [14] J.W. Moon, On the Line-Graph of the Complete Bigraph, Ann. Math. Statist. 34 (1963) 664-667, doi: 10.1214/aoms/1177704179. | Zbl 0138.19204

[014] [15] M.I. Ostrovskii, Minimum congestion spanning trees in planar graphs, Discrete Math. 310 (2010) 1204-1209, doi: 10.1016/j.disc.2009.11.016. | Zbl 1230.05112

[015] [16] M.I. Ostrovskii, Minimal congestion trees, Discrete Math. 285 (2004) 219-226, doi: 10.1016/j.disc.2004.02.009. | Zbl 1051.05032

[016] [17] Y. Otachi, H.L. Bodlaender and E.J. van Leeuwen, Complexity Results for the Spanning Tree Congestion Problem, in: WG 2010, 6410 Lecture Notes in Comput. Sci. (Springer-Verlag, 2010) 3-14. | Zbl 1308.68067

[017] [18] S. Simonson, A variation on the min cut linear arrangement problem, Math. Syst. Theory 20 (1987) 235-252, doi: 10.1007/BF01692067. | Zbl 0643.68094