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Sally Cockburn

Abstract

A graph G is a homomorphic preimage of another graph H, or equivalently G is H-colorable, if there exists a graph homomorphism f : GH. A geometric graph is a simple graph G together with a straight line drawing of G in the plane with the vertices in general position. A geometric homomorphism (respectively, isomorphism) is a graph homomorphism (respectively, isomorphism) that preserves edge crossings (respectively, and non-crossings). The homomorphism poset ℊ of a graph G is the set of isomorphism classes of geometric realizations of G partially ordered by the existence of injective geometric homomorphisms. A geometric graph is ℋ-colorable if for some ∈ ℋ. In this paper, we provide necessary and sufficient conditions for to be 𝒫n-colorable for n ≥ 2. Along the way, we also provide necessary and sufficient conditions for to be 𝒦2,3-colorable.

Open access

Július Czap, Jakub Przybyło and Erika Škrabuľáková

Abstract

A graph G = (V, E) is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. In this paper, we study bipartite 1-planar graphs with prescribed numbers of vertices in partite sets. Bipartite 1-planar graphs are known to have at most 3n − 8 edges, where n denotes the order of a graph. We show that maximal-size bipartite 1-planar graphs which are almost balanced have not significantly fewer edges than indicated by this upper bound, while the same is not true for unbalanced ones. We prove that the maximal possible size of bipartite 1-planar graphs whose one partite set is much smaller than the other one tends towards 2n rather than 3n. In particular, we prove that if the size of the smaller partite set is sublinear in n, then |E| = (2 + o(1))n, while the same is not true otherwise.

Open access

Yury Metelsky, Kseniya Schemeleva and Frank Werner

R eferences [1] L.W. Beineke, Derived graphs and digraphs , in: Beitrage zur Graphentheorie, H. Sachs, H.-J. Voss, H.-J. Walter (Ed(s)), (Leipzig, Teubner, 1968) 17–33. [2] J.C. Bermond and J.C. Meyer, Graphs representatif des arêtes d’un multigraphe , J. Math. Pures Appl. 52 (1973) 299–308. [3] V. Chvátal and P.L. Hammer, Set-Packing and Threshold Graphs (Comp. Sci. Dept. Univ. of Waterloo, Ontario, 1973). [4] D.G. Degiorgi and K. Simon, A dynamic algorithm for line graph recognition , Lect. Notes in Comput. Sci. 1017 (1995) 37

Open access

Zhangdong Ouyang

R eferences [1] G. Chartrand and L. Lesniak, Graphs and Digraphs, 3rd Edition (Chapman & Hall, New York, 1996). [2] G.L. Chia and C.S. Gan, On crossing numbers of 5 -regular graphs , in: J.-Y. Cai and C.K. Wong (Eds.), Computing and Combinatorics, Lecture Notes in Comput. Sci. 2387 (2002) 230–237. doi:10.1007/3-540-45655-4_26 [3] G.L. Chia and C.S. Gan, Minimal regular graphs with given girths and crossing numbers , Discuss. Math. Graph Theory 24 (2004) 223–237. doi:10.7151/dmgt.1227 [4] Z.D. Ouyang, J. Wang and Y.Q. Huang, The

Open access

Jing Wang, Zhangdong Ouyang and Yuanqiu Huang

R eferences [1] J. Adamsson and R.B. Richter, Arrangements, circular arrangements and the crossing number of C 7 × C n , J. Combin. Theory Ser. B 90 (2004) 21–39. doi:10.1016/j.jctb.2003.05.001 [2] L.W. Beineke and R.D. Ringeisen, On the crossing numbers of products of cycles and graphs of order four , J. Graph Theory 4 (1980) 145–155. doi:10.1002/jgt.3190040203 [3] D. Bokal, On the crossing numbers of Cartesian products with paths , J. Combin. Theory Ser. B 97 (2007) 381–384. doi:10.1016/j.jctb.2006.06.003 [4] D. Bokal, On the

Open access

Adam Naumowicz

Summary

In this paper we account for the formalization of the seven bridges of Königsberg puzzle. The problem originally posed and solved by Euler in 1735 is historically notable for having laid the foundations of graph theory, cf. [7]. Our formalization utilizes a simple set-theoretical graph representation with four distinct sets for the graph’s vertices and another seven sets that represent the edges (bridges). The work appends the article by Nakamura and Rudnicki [10] by introducing the classic example of a graph that does not contain an Eulerian path.

This theorem is item #54 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.

Open access

Terry A. McKee

R eferences [1] A. Brandstädt, V.B. Le and J.P. Spinrad, Graph Classes: A Survey (Society for Industrial and Applied Mathematics, Philadelphia, 1999). doi:10.1137/1.9780898719796 [2] R.C.S. Machado, C.M.H. de Figueiredo and N. Trotignon, Complexity of colouring problems restricted to unichord-free and { square, unichord }- free graphs , Discrete Appl. Math. 164 (2014) 191–199. doi:10.1016/j.dam.2012.02.016 [3] R.C.S. Machado, C.M.H. de Figueiredo and K. Vušković, Chromatic index of graphs with no cycle with a unique chord , Theoret. Comput

Open access

Yang Gao and Heping Zhang

.08.007 [5] C. Godsil and G. Royle, Algebraic Graph Theory (Springer-Verlag, New York, 2001). doi:10.1007/978-1-4613-0163-9 [6] B. Grünbaum and T. Motzkin, The number of hexagons and the simplicity of geodesics on certain polyhedra , Canad. J. Math. 15 (1963) 744–751. doi:10.4153/CJM-1963-071-3 [7] E. Hartung, Fullerenes with complete Clar structure , Discrete Appl. Math. 161 (2013) 2952–2957. doi:10.1016/j.dam.2013.06.009 [8] T. Pisanski and M. Randić, Bridges between geometry and graph theory , in: Geometry at Work: Papers in Applied Geometry Vol

Open access

Zhikui Chang and Liping Yuan

-38. [5] Grünbaum, B., Shephard, G. C., Tilings and Patterns. New York: W. H. Freeman and Company, 1987. [6] A. D. Jumani and T. Zamfirescu, On longest paths in triangular lattice graphs, Util. Math. 89 (2012) 269-273. [7] S. Klavžar, M. Petkov_sek, Graphs with non empty intersection of longest paths, Ars Combin. 29 (1990) 13-52. [8] B. Menke, Ch. Zamfirescu and T. Zamfirescu, Intersections of longest cycles in grid graphs, J. Graph Theory 25 (1997) 37-52. [9] F. Nadeem, A. Shabbir, and T. Zamfirescu