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In this paper, we consider various problems concerning quasi-matchings and semi-matchings in bipartite graphs, which generalize the classical problem of determining a perfect matching in bipartite graphs. We prove a generalization of Hall’s marriage theorem, and present an algorithm that solves the problem of determining a lexicographically minimum -quasi-matching (that is a set of edges in a bipartite graph such that in one set of the bipartition every vertex has at least incident edges from , where is a so-called need mapping, while on the other side of the bipartition the distribution of degrees with respect to is lexicographically minimum). We obtain that finding a lexicographically minimum quasi-matching is equivalent to minimizing any strictly convex function on the degrees of the A-side of a quasi-matching and use this fact to prove a more general statement: the optima of any component-based strictly convex cost function on any subset of -sphere in are precisely the lexicographically minimal elements of this subset. We also present an application in designing optimal CDMA-based wireless sensor networks. 相似文献
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A graph is chromatic-index-critical if it cannot be edge-coloured with Δ colours (with Δ the maximal degree of the graph), and if the removal of any edge decreases its chromatic index. The Critical Graph Conjecture stated that any such graph has odd order. It has been proved false and the smallest known counterexample has order [[18] A.J.W. Hilton, R.J. Wilson, Edge-colorings of graphs: a progress report, in: M.F. Cabobianco, et al. (Eds.), Graph Theory and its Applications: East and West, New York, 1989, pp. 241–249; [31] H.P. Yap, Some topics in graph theory, London Mathematical Society, Lecture Note Series, vol. 108, Cambridge University Press, Cambridge, 1986].
In this paper we show that there are no chromatic-index-critical graphs of order 14. Our result extends that of [[5] G. Brinkmann, E. Steffen, Chromatic-index-critical graphs of orders 11 and 12, European J. Combin. 19 (1998) 889–900] and leaves order 16 as the only case to be checked in order to decide on the minimality of the counterexamples given by Chetwynd and Fiol. In addition we list all nontrivial critical graphs of order 13. 相似文献
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Smole Andreja Jagrič Timotej Bokal Drago 《Central European Journal of Operations Research》2021,29(3):791-808
Central European Journal of Operations Research - Our novel game-theoretic Principal/Two-Agent model ensures that the Principal has a reliable internal signal about the Agents’ invested work... 相似文献
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Drago Bokal 《Journal of Graph Theory》2010,65(2):139-162
?iráň constructed infinite families of k‐crossing‐critical graphs for every k?3 and Kochol constructed such families of simple graphs for every k?2. Richter and Thomassen argued that, for any given k?1 and r?6, there are only finitely many simple k‐crossing‐critical graphs with minimum degree r. Salazar observed that the same argument implies such a conclusion for simple k‐crossing‐critical graphs of prescribed average degree r>6. He established the existence of infinite families of simple k‐crossing‐critical graphs with any prescribed rational average degree r∈[4, 6) for infinitely many k and asked about their existence for r∈(3, 4). The question was partially settled by Pinontoan and Richter, who answered it positively for $r\in(3\frac{1}{2},4)$. The present contribution uses two new constructions of crossing‐critical simple graphs along with the one developed by Pinontoan and Richter to unify these results and to answer Salazar's question by the following statement: there exist infinite families of simple k‐crossing‐critical graphs with any prescribed average degree r∈(3, 6), for any k greater than some lower bound Nr. Moreover, a universal lower bound NI on k applies for rational numbers in any closed interval I?(3, 6). © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 139–162, 2010 相似文献
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Drago Bokal Éva Czabarka László A. Székely Imrich Vrt’o 《Discrete and Computational Geometry》2010,44(2):463-483
There are three general lower bound techniques for the crossing numbers of graphs: the Crossing Lemma, the bisection method
and the embedding method. In this contribution, we present their adaptations to the minor crossing number. Using the adapted
bounds, we improve on the known bounds on the minor crossing number of hypercubes. We also point out relations of the minor
crossing number to string graphs and establish a lower bound for the standard crossing number in terms of Randič index. 相似文献
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Drago Bokal Gasper Fijavz Martin Juvan P. Mark Kayll Bojan Mohar 《Journal of Graph Theory》2004,46(3):227-240
We introduce the circular chromatic number χc of a digraph and establish various basic results. They show that the coloring theory for digraphs is similar to the coloring theory for undirected graphs when independent sets of vertices are replaced by acyclic sets. Since the directed k‐cycle has circular chromatic number k/(k – 1), for k ≥ 2, values of χc between 1 and 2 are possible. We show that in fact, χc takes on all rational values greater than 1. Furthermore, there exist digraphs of arbitrarily large digirth and circular chromatic number. It is NP‐complete to decide if a given digraph has χc at most 2. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 227–240, 2004 相似文献
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Consider a graph G with a minimal edge cut F and let G1, G2 be the two (augmented) components of G−F. A long-open question asks under which conditions the crossing number of G is (greater than or) equal to the sum of the crossing numbers of G1 and G2—which would allow us to consider those graphs separately. It is known that crossing number is additive for |F|∈{0,1,2} and that there exist graphs violating this property with |F|≥4. In this paper, we show that crossing number is additive for |F|=3, thus closing the final gap in the question. 相似文献
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Drago Bokal 《Journal of Graph Theory》2007,56(4):287-300
Zip product was recently used in a note establishing the crossing number of the Cartesian product K1,n □ Pm. In this article, we further investigate the relations of this graph operation with the crossing numbers of graphs. First, we use a refining of the embedding method bound for crossing numbers to weaken the connectivity condition under which the crossing number is additive for the zip product. Next, we deduce a general theorem for bounding the crossing numbers of (capped) Cartesian product of graphs with trees, which yields exact results under certain symmetry conditions. We apply this theorem to obtain exact and approximate results on crossing numbers of Cartesian product of various graphs with trees. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 287–300, 2007 相似文献
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