# Zero-sum flow number of octagonal grid and generalized prism

### Abstract

A zero-sum flow is an assignment of nonzero integers to the edges such that the sum of the values of all edges incident with each vertex is zero, and we call it a zero-sum k-flow if the absolute values of edges are less than k. We recall the zero-sum flow number of G as the least integer k for which G admitting a zero sum k-flow. In this paper we gave complete zero-sum flow and zero sum numbers for Octagonal Grid and Generalized Prism.### References

S. Akbari, A. Daemi, O. Hatami, A. Javanmard, and A. Mehrabian, “Zero-sum flows in regular graphs”, Graphs and combinatorics, vol. 26, no. 5, pp. 603–615, Apr. 2010, doi: 10.1007/s00373-010-0946-5.

S. Akbari, N. Ghareghani, G. Khosrovshahi, and A. Mahmoody, “On zero-sum 6-flows of graphs,” Linear algebra and its applications, vol. 430, no. 11-12, pp. 3047–3052, Jun. 2009, doi: 10.1016/j.laa.2009.01.027.

M. Bača and M. Siddiqui, “Total edge irregularity strength of generalized prism”, Applied mathematics and computation, vol. 235, pp. 168–173, May 2014, doi: 10.1016/j.amc.2014.03.001.

S. Chiang and J. Yan, “On L(d,1)-labeling of Cartesian product of a cycle and a path,” Discrete applied mathematics, vol. 156, no. 15, pp. 2867–2881, Aug. 2008, doi: 10.1016/j.dam.2007.11.019.

C. Chang, M. Chia, C. Hsu, D. Kuo, L. Lai, and F. Wang, “Global defensive alliances of trees and Cartesian product of paths and cycles,” Discrete applied mathematics, vol. 160, no. 4-5, pp. 479–487, Mar. 2012., doi: 10.1016/j.dam.2011.11.004.

D. Li and M. Liu, “Incidence colorings of Cartesian products of graphs over path and cycles”, Advances in mathematics, vol. 40, no. 6, pp. 697-708, 2011.

S. Gravier and M. Mollard, “On domination numbers of Cartesian product of paths”, Discrete applied mathematics, vol. 80, no. 2-3. pp. 247-250, Dec. 1997, doi: 10.1016/S0166-218X(97)00091-7.

W. Imrich and S. Klavžar, Product graphs: structure and recognition. New York, NY: Wiley, 2000.

F. Jaeger, “Flows and generalized coloring theorems in graphs”, Journal of combinatorial theory, series B, vol. 26, no. 2, pp. 205-216, Apr. 1979, doi: 10.1016/0095-8956(79)90057-1.

D. Kuo, J. Yan, “On L (2, 1) labelings of Cartesian products of paths and cycles”, Discrete mathematics, vol. 283, no. 1-3, pp. 137-144, Jun. 2004, doi: 10.1016/j.disc.2003.11.009.

Y. Lai, C. Tian, T. Ko, “Edge addition number of Cartesian product of paths and cycles”, Electronic notes in discrete mathematics, vol. 22, pp. 439-444, Oct. 2005, doi: 0.1016/j.endm.2005.06.062.

D. Rall, “Total domination in categorical products of graphs”, Discussiones mathematicae graph theory, vol. 25, no. 1-2, pp. 35-44, 2005, doi: 10.7151/dmgt.1257.

P. Seymour, “Nowhere-zero 6-flows”, Journal of combinatorial theory, series B, vol. 30, no. 2, pp. 130-135, Apr. 1981, doi: 10.1016/0095-8956(81)90058-7.

M. Siddiqui, M. Miller, J. Ryan, “Total edge irregularity strength of octagonal grid graph”, Utilitas mathematica, vol. 103, pp. 277-287, 2017.

C. Tardif and D. Wehlau, “Chromatic numbers of products of graphs: the directed and undirected versions of the Poljak-Rödl function”, Journal of graph theory, vol. 51, no. 1, pp. 33-36, Aug. 2005, doi: 10.1002/jgt.20117.

W. Tutte, “A contribution to the theory of chromatic polynomials”, Canadian journal of mathematics, vol. 6, pp. 80-91, 1954, doi: 10.4153/CJM-1954-010-9.

T. Wang and S. Hu, “Zero-sum flow numbers of regular graphs,” in Frontiers in algorithmics and algorithmic aspects in information and management, J. Snoeyink, K. Su, and L. Wang, Eds. Berlin: Springer, 2012, pp. 269–278, doi: 10.1007/978-3-642-29700-7_25.

T. Wang and S. Hu, “Constant sum flows in regular graphs”, in Frontiers in algorithmics and algorithmic aspects in information and management, M. Atallah, X. Li, B. Zhu, Eds. Berlin: Springer, 2011, pp. 168-175, doi: 10.1007/978-3-642-21204-8_20.

T. Wang, S. Hu and G. Zhang, “Zero-sum flow numbers of triangular grids”, in Frontiers in algorithmics and algorithmic aspects in information and management, J. Chen, J. Hopcroft, J. Wang, Eds. Cham: Sprimger, 2014, pp. 264-275, doi: 10.1007/978-3-319-08016-1_24.

T. Wang, G. Zhang, “Zero-sum flow numbers of hexagonal grids”, in Frontiers in algorithmics and algorithmic aspects in information and management, M. Fellows, X. Tan, B. Zhu, Eds. Berlin; Springer, 2013, pp. 339-349, doi: 10.1007/978-3-642-38756-2_34

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Copyright (c) 2019 Muhammad Naeem, Muhammad Imran, Sarfraz Ahmad, Muhammad Kamran Siddiqui

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