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Given a graph sequence denote by T3(Gn) the number of monochromatic triangles in a uniformly random coloring of the vertices of Gn with colors. In this paper we prove a central limit theorem (CLT) for T3(Gn) with explicit error rates, using a quantitative version of the martingale CLT. We then relate this error term to the well-known fourth-moment phenomenon, which, interestingly, holds only when the number of colors satisfies . We also show that the convergence of the fourth moment is necessary to obtain a Gaussian limit for any , which, together with the above result, implies that the fourth-moment condition characterizes the limiting normal distribution of T3(Gn), whenever . Finally, to illustrate the promise of our approach, we include an alternative proof of the CLT for the number of monochromatic edges, which provides quantitative rates for the results obtained in [7].  相似文献   
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A proper edge coloring of a graph G is acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G, denoted by χ a(G), is the least number of colors such that G has an acyclic edge coloring. A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that χ a(G) ≤Δ(G) + 22, if G is a triangle-free 1-planar graph.  相似文献   
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A cross‐free set of size m in a Steiner triple system is three pairwise disjoint m‐element subsets such that no intersects all the three ‐s. We conjecture that for every admissible n there is an STS(n) with a cross‐free set of size which if true, is best possible. We prove this conjecture for the case , constructing an STS containing a cross‐free set of size 6k. We note that some of the 3‐bichromatic STSs, constructed by Colbourn, Dinitz, and Rosa, have cross‐free sets of size close to 6k (but cannot have size exactly 6k). The constructed STS shows that equality is possible for in the following result: in every 3‐coloring of the blocks of any Steiner triple system STS(n) there is a monochromatic connected component of size at least (we conjecture that equality holds for every admissible n). The analog problem can be asked for r‐colorings as well, if and is a prime power, we show that the answer is the same as in case of complete graphs: in every r‐coloring of the blocks of any STS(n), there is a monochromatic connected component with at least points, and this is sharp for infinitely many n.  相似文献   
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We classify the trees on n vertices with the maximum and the minimum number of certain generalized colorings, including conflict-free, odd, non-monochromatic, star, and star rainbow vertex colorings. We also extend a result of Cutler and Radcliffe on the maximum and minimum number of existence homomorphisms from a tree to a completely looped graph on q vertices.  相似文献   
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Consider a simple graph and its proper edge coloring c with the elements of the set . We say that c is neighbor set distinguishing (or adjacent strong) if for every edge , the set of colors incident with u is distinct from the set of colors incident with v. Let us then consider a stronger requirement and suppose we wish to distinguishing adjacent vertices by sums of their incident colors. In both problems the challenging conjectures presume that such colorings exist for any graph G containing no isolated edges if only . We prove that in both problems is sufficient. The proof is based on the Combinatorial Nullstellensatz, applied in the “sum environment.” In fact the identical bound also holds if we use any set of k real numbers instead of as edge colors, and the same is true in list versions of the both concepts. In particular, we therefore obtain that lists of length ( in fact) are sufficient for planar graphs.  相似文献   
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An intervalt-coloring of a multigraph G is a proper edge coloring with colors 1,,t such that the colors of the edges incident with every vertex of G are colored by consecutive colors. A cyclic intervalt-coloring of a multigraph G is a proper edge coloring with colors 1,,t such that the colors of the edges incident with every vertex of G are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. Denote by w(G) (wc(G)) and W(G) (Wc(G)) the minimum and maximum number of colors in a (cyclic) interval coloring of a multigraph G, respectively. We present some new sharp bounds on w(G) and W(G) for multigraphs G satisfying various conditions. In particular, we show that if G is a 2-connected multigraph with an interval coloring, then W(G)1+|V(G)|2(Δ(G)?1). We also give several results towards the general conjecture that Wc(G)|V(G)| for any triangle-free graph G with a cyclic interval coloring; we establish that approximate versions of this conjecture hold for several families of graphs, and we prove that the conjecture is true for graphs with maximum degree at most 4.  相似文献   
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本文利用含参变量广义积分计算中常用的求导和变量替换等技巧等给出了傅里叶变换中一个重要的含参变量广义积分的计算方法.  相似文献   
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