Abstract
The fractional matching preclusion number of a graph G, denoted by fmp(G), is the minimum number of edges whose deletion results in a graph with no fractional perfect matchings. Let \(G_{k,n}\) be the complete n-balanced k-partite graph, whose vertex set can be partitioned into k parts, each has n vertices and whose edge set contains all edges between two distinct parts. In this paper, we prove that if \(k=3\) or 5 and \(n=1\), then \(fmp(G_{k,n})=\delta (G_{k,n})-1\); otherwise \(fmp(G_{k,n})=\delta (G_{k,n})\).
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References
Brigham RC, Harary F, Violin EC, Yellen J (2005) Perfect-matching preclusion. Congr. Numer. 174:185–192
Chen G, Jacobson MS (1997) Degree Sum conditions for Hamiltonicity on k-Partite Graphs. Graphs Comb. 13:325–343
Cheng E, Connolly R, Melekian C (2015) Matching preclusion and conditional matching preclusion problems for the folded Petersen cube. Theor. Comput. Sci. 576:30–44
Cheng E, Hu P, Jia R, Lipták L (2012) Matching preclusion and conditional matching preclusion for bipartite interconnection networks I: Sufficient conditions. Networks 59:349–356
Cheng E, Hu P, Jia R, Lipták L (2012) Matching preclusion and conditional matching preclusion problems for bipartite interconnection networks II: Cayley graphs generated by transposition trees and hyperstars. Networks 59:357–364
Cheng E, Hu P, Jia R, Lipták L, Scholten B, Voss J (2014) Matching preclusion and conditional matching preclusion for pancake and burnt pancake graphs. Int. J. Parallel Emerg. Distrib. Syst. 29:499–512
Cheng E, Lipman MJ, Lipták L (2012) Matching preclusion and conditional matching preclusion for regular interconnection networks. Discrete Appl. Math. 160:1936–1954
Cheng E, Lipták L (2012) Matching preclusion and conditional matching preclusion problems for tori and related Cartersian products. Discrete Appl. Math. 160:1699–1716
Li Q, He J, Zhang H (2016) Matching preclusion for vertex-transitive networks. Discrete Appl. Math. 207:90–98
Li Q, Shiu WC, Yao H (2015) Matching preclusion for cube-connected cycles. Discrete Appl Math. 190(191):118–126
Lin R, Zhang H (2016) Maximally matched and super matched regular graphs. Int. J. Comput. Math. Comput. Syst. Theory 1:74–84
Lin R, Zhang H (2017) Matching preclusion and conditional edge-fault Hamiltonicity of binary de Bruijn graphs. Discrete Appl. Math. 233:104–117
Lin R, Zhang H, Zhao W (2019) Matching preclusion for direct product of regular graphs. Discrete Appl. Math. 277:221–230
Liu Y, Liu W (2017) Fractional matching preclusion of graphs. J. Comb. Optim. 34:522–533
Lv H, Li X, Zhang H (2012) Matching preclusion for balanced hypercubes. Theor. Comput. Sci. 84:109–136
Ma T, Mao Y, Cheng E, Wang J (2019) Fractional matching preclusion for arrangement graphs. Discrete Appl. Math. 270:181–189
Ma T, Mao Y, Cheng E, Wang J (2018) Fractional matching preclusion for \((n, k)\)-star graphs. Parallel Process. Lett. 28(4):1850017
Wang J (2020) Fractional matching preclusion of product networks. Theor. Comput. Sci. 846:75–81
Wang Z, Melekian C, Cheng E, Mao Y (2019) Matching preclusion number in product graphs. Theor. Comput. Sci. 755:38–47
Wang S, Wang R, Lin S, Li J (2010) Matching preclusion for \(k\)-ary \(n\)-cubes. Discrete Appl. Math. 158:2066–2070
Zhang S, Liu H, Li D, Hu X (2019) Fractional matching preclusion of the restricted HL-graphs. J. Comb. Optim. 38:1143–1154
Funding
Mei Lu is supported by the National Natural Science Foundation of China (Grant 11771247 and 11971158); Yi Zhang is supported by the National Natural Science Foundation of China (Grant 11901048 and 12071002).
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Luan, Y., Lu, M. & Zhang, Y. The fractional matching preclusion number of complete n-balanced k-partite graphs. J Comb Optim 44, 1323–1329 (2022). https://doi.org/10.1007/s10878-022-00888-5
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DOI: https://doi.org/10.1007/s10878-022-00888-5