A Note on n-Critical Bipartite Graphs and Its Application

Liu, Yueping; Nie, Zhe

doi:10.1007/978-3-642-02026-1_26

Cited by 3 publications

(8 citation statements)

References 7 publications

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“…A.1 2-extendibility of G 1 Let E 0 = {e 0 , e 1 } be an edge pair in G 1 and let G ′ 1 = G 1 − {(0, 8), (2,4), (1,9), (3, 5)}. Then G ′ 1 is isomorphic to H B 3,10 defined in Section 2, which is 2-extendable.…”

Section: Final Remarksmentioning

confidence: 99%

“…Then G ′ 1 is isomorphic to H B 3,10 defined in Section 2, which is 2-extendable. Hence, if E 0 ∩{(0, 8), (2,4), (1,9), (3, 5)} = ∅, then E 0 is contained in a perfect matching of G ′ 1 , which is also a perfect matching of G 1 . If E 0 ⊂ {(0, 8), (2,4), (1,9), (3, 5)}, then E 0 is contained in the perfect matching {(0, 8), (2,4), (1,9), (3,5), (6, 7)}.…”

Section: Final Remarksmentioning

confidence: 99%

“…What left is the case that |E 0 ∩ {(0, 8), (2,4), (1,9), (3, 5)}| = 1. By symmetry, we only need to verify that every such edge pair containing (0, 8) or (2,4) is contained in a perfect matching of G 1 .…”

Section: Final Remarksmentioning

confidence: 99%

“…Let C 1 = (0, 1, 2, 3, 0) and C 2 = (4, 5, 6,7,8,9,10,11,12,13,14,15,4). The analysis is analog to that in Section A.3.…”

mentioning

confidence: 99%

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Minimum Size of n-Factor-Critical Graphs and k-Extendable Graphs

Zhang

Lou

et al. 2011

Graphs and Combinatorics

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Section: Final Remarksmentioning

confidence: 99%

Section: Final Remarksmentioning

confidence: 99%

Section: Final Remarksmentioning

confidence: 99%

“…Let C 1 = (0, 1, 2, 3, 0) and C 2 = (4, 5, 6,7,8,9,10,11,12,13,14,15,4). The analysis is analog to that in Section A.3.…”

mentioning

confidence: 99%

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Minimum Size of n-Factor-Critical Graphs and k-Extendable Graphs

Zhang

Lou

et al. 2011

Graphs and Combinatorics

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“…The reader may trace the important developments on this field by referring to the surveys [124], [125] and [126] by Plummer, as well as Chapter 6 and Chapter 7 of the book [141] by Yu and Liu. Furthermore, a good description of the application of extendibility and factor-criticality in job assignment can be found in [104].…”

Section: Minimum Size Of N-factor-critical and K-extendable Graphsmentioning

confidence: 99%

Algorithmic and structural aspects of graph partitioning and related problems

Zhang

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Chapter 1 1.1.1 Monochromatic clique and rainbow cycle partitions In Chapter 2, we study graph problems related to coloring and partitioning restricted to graphs avoiding certain fixed induced subgraphs. The research on problems regarding coloring and partitioning has a relatively long history, and many important and impressive results have been obtained (See, e.g., Erdős et al. [41], Gyárfás and Simonyi [67], Gyárfás et al. [66], Alon et al. [6], Brualdi and Hollingsworth [30], Alon and Gutin [7], Feder et al. [43], Feder and Motwani [44], Suzuki [130], Akbari and Alipour [3], and Gourvès et al. [60]). Several variations of such problems, and in particular their computational complexity, have been well-studied as well. MacGillivray and Yu [110] studied a general graph partitioning problem including graph coloring, homomorphism to H, conditional coloring, contractibility to H, and partition into cliques as special cases, and investigated its complexity. Yegnanarayanan [139] considered three coloring parameters of a graph G in connection with the computational complexity, partitions, algebra, projective plane geometry and analysis. For more general coloring and partitioning problems, the reader could refer to Garey and Johnson [54], and Kano and Li [86]. Let K − 4 denote the graph obtained by deleting one edge from a K 4. A graph G is said to be K − 4-free if it does not contain K − 4 as an induced subgraph. And a graph G is called monochromatic-K − 4-free if any monochromatic subgraph of G does not contain a K − 4 as an induced subgraph. Note that the properties of being K − 4-free and monochromatic-K − 4-free do not imply each other. For example, a K 4 with one edge colored 1 and the others colored 2 is K − 4free, but not monochromatic-K − 4-free. However, a monochromatic cycle on 4 Motivated by the vast existing literature on the on-line versions of coloring problems, we also study the on-line version of injective coloring. The injective coloring problem gets more complicated in the on-line situation. In this case, vertices of a graph are presented one at a time, and the algorithm has to assign a color irrevocably to a vertex as it comes in. The procedure depends only on the knowledge of the subgraph that has been revealed so When the hypergraph is 2-uniform (a standard graph), MHC-LU is known as the maximum cut with limited unbalance problem (MC-LU). Galbiati and Maffioli [53] developed polynomial time randomized approximation algorithms with nontrivial performance guarantees for MC-LU. The well-known max cut problem is equivalent to MHC-LU with u = m and |S j | = 2 for all j. Goemans and Williamson [57], in a major breakthrough, used semidefinite programming relaxation and hyperplane rounding to obtain an approximation algorithm for the Max Cut problem with expected performance guarantee 0.87856. This well-known algorithmic paradigm, with more sophisticated techniques, has been applied to many previously studied problems [49, 50, 52, 53, 70, 71, 138, 143]. When u = 0 and |S j | = 2 for all j, MHC-LU is known ...

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Minimum size of n-factor-critical graphs and k-extendable graphs

Zhang

Lou

et al. 2017

Preprint

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A Note on n-Critical Bipartite Graphs and Its Application

Cited by 3 publications

References 7 publications

Minimum Size of n-Factor-Critical Graphs and k-Extendable Graphs

Minimum Size of n-Factor-Critical Graphs and k-Extendable Graphs

Algorithmic and structural aspects of graph partitioning and related problems

Minimum size of n-factor-critical graphs and k-extendable graphs

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