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Fan Wang and Weisheng Zhao

(2009) 1711–1713. doi:10.1016/j.disc.2008.02.013 [9] L. Gros, Théorie du Baguenodier (Aimé Vingtrinier, Lyon, 1872). [10] G. Kreweras, Matchings and Hamiltonian cycles on hypercubes , Bull. Inst. Combin. Appl. 16 (1996) 87–91. [11] F. Ruskey and C. Savage, Hamilton cycles that extend transposition matchings in Cayley graphs of S n , SIAM J. Discrete Math. 6 (1993) 152–166. doi:10.1137/0406012 [12] J. Vandenbussche and D. West, Matching extendability in hypercubes , SIAM J. Discrete Math. 23 (2009) 1539–1547. doi:10

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Donglin Liu, Chunxiang Wang and Shaohui Wang

path joining u and v is called a uv -path. A path P in graph G is call a Hamiltonian path if V ( P ) = V ( G ). A cycle is a path such that the first vertex is the same as the last one. A cycle is also denoted by x 1 x 2 · · · x l x 1 . Length of a cycle is the number of edges in it. An ‘ -cycle is a cycle of length l . Let c ( G ) = max{l : ‘ -cycle of G} . A cycle of G is called Hamiltonian cycle if its length is |V ( G ) | . A graph G is Hamiltonian if G contains a Hamiltonian cycle. A graph G is Hamiltonian-connected if any

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Wojciech Wideł

[8] B. Ning, Pairs of Fan-type heavy subgraphs for pancyclicity of 2- connected graphs , Australas. J. Combin. 58 (2014) 127–136. [9] B. Ning and S. Zhang, Ore- and Fan-type heavy subgraphs for Hamiltonicity of 2- connected graphs , Discrete Math. 313 (2013) 1715–1725. doi:10.1016/j.disc.2013.04.023 [10] E.F. Schmeichel and S.L. Hakimi, A cycle structure theorem for Hamiltonian graphs , J. Combin. Theory Ser. B 45 (1988) 99–107. doi:10.1016/0095-8956(88)90058-5

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Pawaton Kaemawichanurat

. Blitch, Domination critical graphs , J. Combin. Theory Ser. B 34 (1983) 65–76. doi:10.1016/0095-8956(83)90007-2 [12] D.W. Thacker, Double Domination Edge Critical Graph, Master Thesis (East Tennessee State University, 2006). [13] F. Tian, B. Wei and L. Zhang, Hamiltonicity in 3 -domination critical graphs with α = δ + 2, Discrete Appl. Math. 92 (1999) 57–70. doi:10.1016/S0166-218X(98)00149-8 [14] H.C. Wang and L.Y. Kang, Matching properties in double domination edge critical graphs , Discrete Math. Algorithms Appl. 2 (2010) 151–160. doi

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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/.

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Binlong Li and Bo Ning

(1952) 69–81. doi:10.1112/plms/s3-2.1.69 [12] D. Duffus, M. Jacboson and R.J. Gould, Forbidden subgraphs and the Hamiltonian theme , in: The Theory and Applications of Graphs (Wiley, New York, 1981) 297–316. [13] G. Fan, New sufficient conditions for cycles in graphs , J. Combin. Theory Ser. B 37 (1984) 221–227. doi:10.1016/0095-8956(84)90054-6 [14] R.J. Faudree and R.J. Gould, Characterizing forbidden pairs for hamiltonian properties , Discrete Math. 173 (1997) 45–60. doi:10.1016/S0012-365X(96)00147-1 [15] R.J. Faudree, R.J. Gould, Z

Open access

Yutaro Sakamoto

domination number of generalized Petersen graphs P(n, 2), Discrete Math. 309 (2009) 2445-2451. doi: 10.1016/j.disc.2008.05.026 [5] R.M. Karp, Reducibility among combinatorial problems in: R.E. Miller and J.W. Thatcher (Eds.), (Plenum Press, New York, 1972) 85-103 doi: 10.1007/978-1-4684-2001-2 9. [6] K. Kutnar and P. Petecki, On automorphisms and structural properties of double generalized Petersen graphs, Discrete Math. 339 (2016) 2861-2870. doi: 10.1016/j.disc.2016.05.032 [7] M.E. Watkins, A theorem on Tait colorings with

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Charles Brian Crane

P. Zhang, Graphs and Digraphs, 5 th Edition (Chapman and Hall/CRC, Boca Raton, FL, 2011). [5] C.B. Crane, Generalized pancyclic properties in claw-free graphs , Graphs Combin. 31 (2015) 2149–2158. doi:10.1007/s00373-014-1510-5 [6] Y. Egawa, J. Fujisawa, S. Fujita and K. Ota, On 2 -factors in r-connected { K 1, k , P 4 } -free graphs , Tokyo J. Math. 31 (2008) 415–420. doi:10.3836/tjm/1233844061 [7] R.J. Faudree and R.J. Gould, Characterizing forbidden pairs for Hamiltonian properties , Discrete Math. 173 (1997) 45–60. doi:10.1016/S

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Juan José Montellano-Ballesteros and Anahy Santiago Arguello

R eferences [1] N. Alon and Y. Roichman, Random Cayley graphs and expanders , Random Structures Algorithms 5 (1994) 271–284. doi:10.1002/rsa.3240050203 [2] J.A. Bondy and U.S.R. Murty, Graph Theory (Springer, New York, 2008). [3] J. Bourgain and A. Gamburd, Uniform expansion bounds for Cayley graphs of SL 2 (ℱ p ), Ann. of Math. 167 (2008) 625–642. doi:10.4007/annals.2008.167.625 [4] C.C. Chen and N. Quimpo, On strongly hamiltonian abelian group graphs , Combin. Math. VIII (Geelong, 1980) Lecture Notes in Math. 884 (Springer

Open access

R. Çolak, Y. Altin and M. Et

References 1. Banach, S. - Theorie des Operations Lineaires, Chelsea Publishing Co., New York, 1955. 2. Colak, R.; Bektaş, C.A. - λ-statistical convergence of order α, Acta Math. Sci. Ser. B Engl. Ed., 31 (2011), 953-959. 3. Colak, R. - Statistical convergence of order α, Modern Methods in Analysis and Its Applications, NewDelhi, India, Anamaya Pub, 2010, 121-129. 4. Colak, R.; Cakar, O. - Banach limits and related matrix transformations, Studia Sci. Math. Hungar., 24 (1989), 429-436. 5. Connor, J.S. - The statistical and strong p