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引用次数: 0
摘要
从 maxcut 的概念出发，研究有符号图的挫折指数是有符号图理论的基本问题之一。最近，两位作者发起了对临界挫折有符号图的研究。这是一种有符号图，其沮度指数随着任何边的移除而减小。本研究的重点是临界有符号图，这些图不是临界受挫有符号图（即不可分解有符号图）的边缘相交的联合体，也不是通过细分从其他临界受挫有符号图建立起来的。我们猜想，对于任何给定的 k，只有有限多个此类临界 k 受挫有符号图。为了支持这一猜想，我们证明了只有两个此类临界 3 受挫有符号图不存在一对边缘相接的负循环。同样，我们证明了正好有十个临界三挫折有符号平面图既不是可分解的，也不是其他临界三挫折有符号图的细分。我们提出了一种基于两个给定的此类有符号图形构建不可分解的临界受挫有符号图形的方法。我们还证明了不可分解的条件对于我们的猜想是必要的。
Extending the notion of maxcut, the study of the frustration index of signed graphs is one of the basic questions in the theory of signed graphs. Recently two of the authors initiated the study of critically frustrated signed graphs. That is a signed graph whose frustration index decreases with the removal of any edge. The main focus of this study is on critical signed graphs which are not edge-disjoint unions of critically frustrated signed graphs (namely indecomposable signed graphs) and which are not built from other critically frustrated signed graphs by subdivision. We conjecture that for any given k there are only finitely many critically k-frustrated signed graphs of this kind.
Providing support for this conjecture we show that there are only two of such critically 3-frustrated signed graphs where there is no pair of edge-disjoint negative cycles. Similarly, we show that there are exactly ten critically 3-frustrated signed planar graphs that are neither decomposable nor subdivisions of other critically frustrated signed graphs. We present a method for building indecomposable critically frustrated signed graphs based on two given such signed graphs. We also show that the condition of being indecomposable is necessary for our conjecture.
期刊介绍：
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.
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