Báo cáo toán học: "Colouring 4-cycle systems with speciﬁed block colour

Báo cáo toán học: "Colouring 4-cycle systems with speciﬁed block colour patterns: the case of embedding P3-designs" ppsx

... that the existence of a 4-cycle system of order n having an m-colouring of type bd, implies the one of a 4-cycle system of order n + 8 having an (m + 1)-colouring of type bd. ✷4 2-Colouring of ... ,n+34} thereis a 4-cycle system of order n with a proper m-colouring of type bf .Proof.Thecasesm =3andm =n+34are proved by using Theorem 2.3 and Theorem2.4 respectively.Starting from the 3-coloured ... which only speciﬁed block colouring patterns areallowed. In this paper we want to consider strict colouring in the sense of Voloshin of 4-cycle systems in which only speciﬁed block colouring...

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Báo cáo toán học: "On edge colorings with at least q colors in every subset of p vertices" ppsx

... consider the 2ai+1vertices in V (Mi+1) and the edges of Miincident to these vertices. We have four cases. Case 1: If 2ai+1= ai(so we have the ﬁrst case in (3)), then this is Mi. Case ... concepts see the monograph of Bollob´as [1]. V (G)andE(G)denotethevertex-set and the edge-set of the graph G. Knis the complete graph on n vertices. Inthis paper log n denotes the base 2 logarithm. ... bj−1(unless i=0). Case 3: If 2ai+1= ai− 1 (third case in (3)) and there is no marked vertex at the moment, then to get Miwe add one arbitrary edge of Mito these 2ai+1edges. Case 4: Finally,...

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Báo cáo toán học: "On some continuation problems which are closely related to the theory of operators in spaces $Pi_chi$. IV " pps

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Báo cáo toán học: "A simple Havel–Hakimi type algorithm to realize graphical degree sequences of directed graphs" ppsx

... step.After the last step we ﬁnd that the sequence reduced by C is graphical. Proof of Theorem 2: We can easily achieve now the required graphical realization of (d+, d−) if we use Lemma 6 with the ... of the complete graph G, but their proof wasindependent of Tutte’s proof and they referred to Havel’s theorem.In 1962 S.L. Hakimi studied the degree sequence problem in undirected graphs with multiple ... In this case there will be a set D of (di)graphs satisfying the existing data, and one can be faced with: (i) ﬁnding a typical element of the class D,(ii) or generating all elements of the class...