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P. Malakooti Rad, S. Yassemi, Sh. Ghalandarzadeh and P. Safari

) 755–779. [14] Sh. Ghalandarzadeh, S. Shirinkam, P. Malakooti Rad, Annihilator Ideal-Based Zero-Divisor Graphs Over Multiplication Modules, Communications in Algebra . 41 (2013) 1134–1148. [15] D.C. Lu, T.S. Wu, On bipartite zero-divisor graphs, Discrete Math . 309 (2009) 755-762. [16] P. Malakooti Rad, Sh. Ghalandarzadeh, S. Shirinkam, On The Torsion Graph and Von Numann Regular Rings, Filomat . 26 (2012) 47–53. [17] S.P. Redmond, The zero-divisor graph of a non-commutative ring, Internat. J. Commutative Rings 1 (2002) 203-211.

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Magda Dettlaff, Joanna Raczek and Jerzy Topp

R eferences [1] H. Aram, S.M. Sheikholeslami and O. Favaron, Domination subdivision numbers of trees , Discrete Math. 309 (2009) 622–628. doi:10.1016/j.disc.2007.12.085 [2] D. Avella-Alaminos, M. Dettlaff, M. Lemańska and R. Zuazua, Total domination multisubdivision number of a graph , Discuss. Math. Graph Theory 35 (2015) 315–327. doi:10.7151/dmgt.1798 [3] S. Benecke and C.M. Mynhardt, Trees with domination subdivision number one , Australas. J. Combin. 42 (2008) 201–209. [4] A. Bhattacharya and G.R. Vijayakumar, Effect of edge

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Yelena Mandelshtam

Abstract

In this paper we study graphs defined by pattern-avoiding words. Word-representable graphs have been studied extensively following their introduction in 2004 and are the subject of a book published by Kitaev and Lozin in 2015. Recently there has been interest in studying graphs represented by pattern-avoiding words. In particular, in 2016, Gao, Kitaev, and Zhang investigated 132-representable graphs, that is, word-representable graphs that can be represented by a word which avoids the pattern 132. They proved that all 132-representable graphs are circle graphs and provided examples and properties of 132-representable graphs. They posed several questions, some of which we answer in this paper.

One of our main results is that not all circle graphs are 132-representable, thus proving that 132-representable graphs are a proper subset of circle graphs, a question that was left open in the paper by Gao et al. We show that 123-representable graphs are also a proper subset of circle graphs, and are different from 132-representable graphs. We also study graphs represented by pattern-avoiding 2-uniform words, that is, words in which every letter appears exactly twice.

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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

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Joanna Cyman, Magda Dettlaff, Michael A. Henning, Magdalena Lemańska and Joanna Raczek

. 27 (2007) 166–170, in Chinese. [5] T.C.E. Cheng, L.Y. Kang and C.T. Ng, Paired domination on interval and circular-arc graphs , Discrete Appl. Math. 155 (2007) 2077–2086. doi:10.1016/j.dam.2007.05.011 [6] T.C.E. Cheng, L.Y. Kang and E. Shan, A polynomial-time algorithm for the paired-domination problem on permutation graphs , Discrete Appl. Math. 157 (2009) 262–271. doi:10.1016/j.dam.2008.02.015 [7] W.J. Desormeaux, T.W. Haynes, M.A. Henning and A. Yeo, Total domination in graphs with diameter 2, J. Graph Theory 75 (2014) 91–103. doi:10

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S. Pirzada and M. Imran Bhat

, Resolvability in graphs and metric dimension of a graph, Disc. Appl. Math. , 105 (1-3) (2000), 99–113. [10] B. Corbas and G. D. Williams, Rings of order p 5 . II. Local rings, J. Algebra , 231 (2) (2000), 691–704. [11] R. Diestel, Graph Theory, 4th ed. Vol. 173 of Graduate texts in mathematics, Heidelberg: Springer, 2010. [12] F. Harary and R. A. Melter, On the metric dimension of a graph, Ars Combin. , 2 (1976), 191–195. [13] J. A. Huckaba, Commutative Rings with Zero Divisors , Marcel-Dekker, New York, Basel, 1988. [14] S. Khuller, B

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Dawood Hassanzadeh-Lelekaami and Maryam Karimi

polynomial and power series over commutative rings, Comm. Algebra 33 (2005) 2043-2050. doi: 10.1081/AGB-200063357 [8] F. Azarpanah and M. Motamedi, Zero-divisor graph of C(X), Acta Math. Hungar 108 (2005) 25-36. doi: 10.1007/s10474-005-0205-z [9] A. Azizi, Strongly irreducible ideals, J. Aust. Math. Soc. 84 (2008) 145-154. doi: 10.1017/S1446788708000062 [10] A. Barnard, Multiplication modules, J. Algebra 71 (1981) 174-178. doi: 10.1016/0021-8693(81)90112-5 [11] I. Beck, Coloring of commutative rings, J

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Genealogy, Archive, Image

Interpreting Dynastic History in Western India, c. 1090-2016

Jayasinhji Jhala

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Sebastian Koch

. Formalized Mathematics , 26( 2 ):101–124, 2018. doi:10.2478/forma-2018-0009. [6] Gilbert Lee. Walks in graphs. Formalized Mathematics , 13( 2 ):253–269, 2005. [7] Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics ,6( 3 ):335–338, 1997. [8] Klaus Wagner. Graphentheorie . B.I-Hochschultaschenbücher; 248. Bibliograph. Inst., Mannheim, 1970. ISBN 3-411-00248-4. [9] Robin James Wilson. Introduction to Graph Theory . Oliver & Boyd, Edinburgh, 1972. ISBN 0-05-002534-1.

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Sebastian Koch

, Edinburgh, 1972. ISBN 0-05-002534-1.