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On the Erdős-Gyárfás Conjecture in Claw-Free Graphs

Abstract

The Erdős-Gyárfás conjecture states that every graph with minimum degree at least three has a cycle whose length is a power of 2. Since this conjecture has proven to be far from reach, Hobbs asked if the Erdős-Gyárfás conjecture holds in claw-free graphs. In this paper, we obtain some results on this question, in particular for cubic claw-free graphs

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On the Independence Number of Traceable 2-Connected Claw-Free Graphs

R eferences [1] J.A. Bondy and U.S.R. Murty, Graph Theory, Graduate in Mathematics (Springer, 2008). [2] S. Brandt, O. Favaron and Z. Ryjáček, Closure and stable Hamiltonian properties in claw-free graphs , J. Graph Theory 34 (2000) 30–41. doi:10.1002/(SICI)1097-0118(200005)34:1(30::AID-JGT4)3.0.CO;2-R [3] Z. Chen, Chvátal-Erdőos type conditions for Hamiltonicity of claw-free graphs , Graphs Combin. 32 (2016) 2253–2266. doi:10.1007/s00373-016-1716-9 4. V. Chvátal and P. Erdős, A note on Hamilton circuits , Discrete Math. 2 (1972

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Hamiltonicities of Double Domination Critical and Stable Claw-Free Graphs

R eferences [1] S. Ao, G. MacGillivray and J. Simmons, Hamiltonian properties of independent domination critical graphs , J. Combin. Math. Combin. Comput. 85 (2013) 107–128. [2] J. Brousek, Z. Ryjáček and O. Favaron, Forbidden subgraphs, Hamiltonicity and closure in claw-free graphs , Discrete Math. 196 (1999) 29–50. doi:10.1016/S0012-365X(98)00334-3 [3] V. Chvátal, Tough graphs and Hamiltonian circuits , Discrete Math. 306 (2006) 910–917 (reprinted from Discrete Math. 5 (1973) 215–228). doi:10.1016/j.disc.2006.03.011 [4] M

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On Uniquely Hamiltonian Claw-Free and Triangle-Free Graphs

Abstract

A graph is uniquely Hamiltonian if it contains exactly one Hamiltonian cycle. In this note, we prove that claw-free graphs with minimum degree at least 3 are not uniquely Hamiltonian. We also show that this is best possible by exhibiting uniquely Hamiltonian claw-free graphs with minimum degree 2 and arbitrary maximum degree. Finally, we show that a construction due to Entringer and Swart can be modified to construct triangle-free uniquely Hamiltonian graphs with minimum degree 3.

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The Ryjáček Closure and a Forbidden Subgraph

results in K1 , 3-free graphs, J. Graph Theory 8 (1984) 139-146. doi:10.1002/jgt.3190080116 [5] D.J. Oberly and D.P. Sumner, Every connected, locally connected nontrivial graph with no induced claw is Hamiltonian, J. Graph Theory 3 (1979) 351-356. doi:10.1002/jgt.3190030405 [6] M.D. Plummer and A. Saito, Forbidden subgraphs and bounds on the size of a max- imum matching, J. Graph Theory 50 (2005) 1-12. doi:10.1002/jgt.20087 [7] Z. Ryjáček, On a closure concept in claw-free graphs, J. Combin. Theory Ser. B 70 (1997) 217

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Heavy Subgraphs, Stability and Hamiltonicity

244, Springer, 2008). [5] H.J. Broersma, Z. Ryjáček and I. Schiermeyer, Dirac’s minimum degree condition restricted to claws , Discrete Math. 167/168 (1997) 155–166. doi:10.1016/S0012-365X(96)00224-5 [6] H.J. Broersma and H.J. Veldman, Restrictions on induced subgraphs ensuring hamiltonicity or pancyclicity of K 1,3 -free graphs , in: Contemporary Methods in Graph Theory (BI Wissenschaftsverlag, Mannheim 1990) 181–194. [7] J. Brousek, Minimal 2 -connected non-Hamiltonian claw-free graphs , Discrete Math. 191 (1998) 57–64. doi:10.1016/S0012

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On q-Power Cycles in Cubic Graphs

.S. Nowbandegani, H. Esfandiari, M.H.S. Haghighi and K. Bibak, On the Erdős-Gyárfás conjecture in claw-free graphs , Discuss. Math. Graph Theory 34 (2014) 635–640. doi:10.7151/dmgt.1732 [6] S.E. Shauger, Results on the Erdős-Gyárfás conjecture in K 1, m -free graphs , Congr. Numer. 134 (1998) 61–65. [7] D. West, Erdős-Gyárfás conjecture on 2 -power cycle lengths , Open Problems—Graph Theory and Combinatorics. http://www.math.illinois.edu/~dwest/openp/2powcyc.html

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Forbidden Pairs and (k,m)-Pancyclicity

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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Spanning Trees whose Stems have a Bounded Number of Branch Vertices

) 6146-6148. doi:10.1016/j.disc.2009.04.023 [8] H. Matsuda, K. Ozeki and T. Yamashita, Spanning trees with a bounded number of branch vertices in a claw-free graph, Graphs Combin. 30 (2014) 429-437. doi:10.1007/s00373-012-1277-5 [9] M. Tsugaki and Y. Zhang, Spanning trees whose stems have a few leaves, Ars Com- bin. CXIV (2014) 245-256.

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Some Results on the Independence Polynomial of Unicyclic Graphs

531 (2010) 149–157. doi:10.1090/conm/531/10464 [5] J.I. Brown, C.A. Hickman and R.J. Nowakowski, On the location of roots of independence polynomials , J. Algebraic Combin. 19 (2004) 273–282. doi:10.1023/B:JACO.0000030703.39946.70 [6] M. Chudnovsky and P. Seymour, The roots of the independence polynomial of a claw free graph , J. Combin. Theory Ser. B 97 (2007) 350–357. doi:10.1016/j.jctb.2006.06.001 [7] P. Csikvári and M.R. Oboudi, On the roots of edge cover polynomials of graphs , European J. Combin. 32 (2011) 1407–1416. doi:10.1016/j

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