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Brooks’ theorem is a fundamental result in the theory of graph coloring. Catlin proved the following strengthening of Brooks’ theorem: Let dd be an integer at least 3, and let GG be a graph with maximum degree dd. If GG does not contain Kd+1Kd+1 as a subgraph, then GG has a dd-coloring in which one color class has size α(G)α(G). Here α(G)α(G) denotes the independence number of GG. We give a unified proof of Brooks’ theorem and Catlin’s theorem.  相似文献   

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Let (X,d)(X,d) be a metric space endowed with a graph GG such that the set V(G)V(G) of vertices of GG coincides with XX. We define the notion of GG-Reich type maps and obtain a fixed point theorem for such mappings. This extends and subsumes many recent results which were obtained for other contractive type mappings on ordered metric spaces and for cyclic operators.  相似文献   

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Let us fix a function f(n)=o(nlnn)f(n)=o(nlnn) and real numbers 0≤α<β≤10α<β1. We present a polynomial time algorithm which, given a directed graph GG with nn vertices, decides either that one can add at most βnβn new edges to GG so that GG acquires a Hamiltonian circuit or that one cannot add αnαn or fewer new edges to GG so that GG acquires at least e−f(n)n!ef(n)n! Hamiltonian circuits, or both.  相似文献   

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Let FF be either the real number field RR or the complex number field CC and RPnRPn the real projective space of dimension n. Theorems A and C in Hemmi and Kobayashi (2008) [2] give necessary and sufficient conditions for a given FF-vector bundle over RPnRPn to be stably extendible to RPmRPm for every m?nm?n. In this paper, we simplify the theorems and apply them to the tangent bundle of RPnRPn, its complexification, the normal bundle associated to an immersion of RPnRPn in Rn+rRn+r(r>0)(r>0), and its complexification. Our result for the normal bundle is a generalization of Theorem A in Kobayashi et al. (2000) [8] and that for its complexification is a generalization of Theorem 1 in Kobayashi and Yoshida (2003) [5].  相似文献   

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