Estimating the Minimal Number of Colors in Acyclic <InlineEquation ID="IE1"> <EquationSource Format="MATHTYPE"> <![CDATA[% MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb % qeguuDJXwAKbacfiGae83AaSgaaa!3C30! ]]> </EquationSource> <EquationSource Format="TEX"> <![CDATA[$$k $$]]> </EquationSource> </InlineEquation>-Strong Colorings of Maps on Surfaces
نویسندگان
چکیده
A coloring of the vertices of a graph is called acyclic if the ends of each edge are colored in distinct colors, and there are no two-colored cycles. Suppose each face of rank k , k ≥ 4 , in a map on a surface S is replaced by the clique having the same number of vertices. It is proved in [1] that the resulting pseudograph admits an acyclic coloring with the number of colors depending linearly on N and k . In the present paper we prove a sharper estimate 55(−Nk)4/7 for the number of colors provided that k ≥ 1 and −N ≥ 5k .
منابع مشابه
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We obtain the decomposition of the tensor spacesl⊗k n as a module forsln , find an explicit formula for the multiplicities of its irreducible summands, and (when n ≥ 2k) describe the centralizer algebra C = Endsln (sl⊗k n ) and its representations. The multiplicities of the irreducible summands are derangement numbers in several important instances, and the dimension of C is given by the number...
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