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Light graphs in families of polyhedral graphs with prescribed minimum degree, face size, edge and dual edge weight. (English) Zbl 1222.05217
Summary: A graph $$H$$ is defined to be light in a family $$\mathcal H$$ of graphs if there exists a finite number $$\varphi(H,\mathcal H)$$ such that each $$G \in \mathcal H$$ which contains $$H$$ as a subgraph, contains also a subgraph $$K\cong H$$ such that the $$\varDelta _G(K)\leq \varphi(H,\mathcal H)$$. We study light graphs in families of polyhedral graphs with prescribed minimum vertex degree $$\delta$$, minimum face degree $$\rho$$, minimum edge weight $$w$$ and dual edge weight $$w^{*}$$. For those families, we show that there exists a variety of small light cycles; on the other hand, we also present particular constructions showing that, for certain families, the spectrum of short cycles contains irregularly scattered cycles that are not light.

##### MSC:
 05C75 Structural characterization of families of graphs 05C38 Paths and cycles
##### Keywords:
light graph; polyhedral graph; edge weight
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##### References:
 [1] Borodin, O.V., Joint generalization of the Lebesgue and kotzig theorems on combinatorics of plane maps, Diskret. mat., 3, 4, 24-27, (1991), (in Russian) · Zbl 0742.05034 [2] Borodin, O.V.; Woodall, D., Short cycles of low weight in normal plane maps with minimum degree 5, Discuss. math. graph theory, 16, 159-164, (1998) · Zbl 0927.05069 [3] Fabrici, I.; Hexel, E.; Jendrol’, S.; Walther, H., On vertex-degree restricted paths in polyhedral graphs, Discrete math., 212, 61-73, (2000) · Zbl 0946.05047 [4] Fabrici, I.; Jendrol’, S., Subgraphs with restricted degrees of their vertices in planar 3-connected graphs, Graphs combin., 13, 245-250, (1997) · Zbl 0891.05025 [5] Ferencová, B.; Madaras, T., On the structure of polyhedral graphs with prescribed edge and dual edge weight, Acta univ. M. belii, 12, 13-18, (2005) · Zbl 1103.05026 [6] R. Hajduk, R. Soták, On large light graphs in families of polyhedral graphs, Discrete Math., in press (doi:10.1016/j.disc.2009.03.008) · Zbl 1216.05123 [7] Harant, J.; Jendrol’, S.; Tkáč, M., On 3-connected plane graphs without triangular faces, J. combin. theory ser. B, 77, 150-161, (1999) · Zbl 1027.05030 [8] Jendrol’, S.; Madaras, T., On light subgraphs in plane graphs of minimum degree five, Discuss. math. graph theory, 16, 207-217, (1996) · Zbl 0877.05050 [9] Jendrol’, S.; Madaras, T.; Soták, R.; Tuza, Z., On light cycles in plane triangulations, Discrete math., 197-198, 453-467, (1999) · Zbl 0936.05065 [10] Jendrol’, S.; Owens, P., On light graphs in 3-connected plane graphs without triangular or quadrangular faces, Graphs combin., 17, 4, 659-680, (2001) · Zbl 0988.05031 [11] S. Jendrol’, H.-J. Voss, Light subgraphs of graphs embedded in the plane and in the projective plane — A survey (submitted for publication) · Zbl 1259.05045 [12] P. Hudák, Š. Kocel’ová, T. Madaras, On the structure of plane graphs extremal according to Kotzig theorem, Preprint, Inst. of Algebra MATH-AL-2-2001, TU Dresden [13] Kotzig, A., Contribution to the theory of Eulerian polyhedra, Mat. čas. SAV (math. slovaca), 5, 101-113, (1955) [14] Lebesgue, H., Quelques consequences simples de la formule d’euler, J. math. pures appl., 19, 19-43, (1940) · JFM 66.0736.03 [15] Madaras, T., On the structure of plane graphs with minimum face size 5, Discuss. math. graph theory, 24, 3, 403-411, (2004) · Zbl 1063.05038 [16] Madaras, T.; Škrekovski, R., Heavy paths, light stars and big melons, Discrete math., 286, 115-131, (2004) · Zbl 1053.05035 [17] Madaras, T.; Škrekovski, R., Lightness, heaviness and gravity, Discrete math., 307, 939-951, (2007) · Zbl 1111.05027 [18] Madaras, T.; Škrekovski, R.; Voss, H.-J., The 7-cycle $$C_7$$ is light in the family of all plane graphs with minimum degree 5, Discrete math., 307, 1430-1435, (2007) · Zbl 1115.05022 [19] Mohar, B.; Škrekovski, R.; Voss, H.-J., Light subgraphs in planar graphs of minimum degree 4 and edge degree 9, J. graph theory, 44, 261-295, (2003) · Zbl 1031.05075 [20] Robertson, N.; Sanders, D.P.; Seymour, P.D.; Thomas, R., The four-colour theorem, J. combin. theory ser. B, 70, 2-44, (1999) · Zbl 0883.05056
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