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Unavoidable set of face types for planar maps

100%
Horňák M., Jendrol S.
Discussiones Mathematicae Graph Theory
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1996
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tom 16
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nr 2
123-141
EN
The type of a face f of a planar map is a sequence of degrees of vertices of f as they are encountered when traversing the boundary of f. A set 𝒯 of face types is found such that in any normal planar map there is a face with type from 𝒯. The set 𝒯 has four infinite series of types as, in a certain sense, the minimum possible number. An analogous result is applied to obtain new upper bounds for the cyclic chromatic number of 3-connected planar maps.
2
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Localization of jumps of the point-distinguishing chromatic index of $K_{n,n}$

100%
Horňák M., Soták R.
Discussiones Mathematicae Graph Theory
|
1997
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tom 17
|
nr 2
243-251
EN
The point-distinguishing chromatic index of a graph represents the minimum number of colours in its edge colouring such that each vertex is distinguished by the set of colours of edges incident with it. Asymptotic information on jumps of the point-distinguishing chromatic index of $K_{n,n}$ is found.
3
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On-line ranking number for cycles and paths

100%
Bruoth E., Horňák M.
Discussiones Mathematicae Graph Theory
|
1999
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tom 19
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nr 2
175-197
EN
A k-ranking of a graph G is a colouring φ:V(G) → {1,...,k} such that any path in G with endvertices x,y fulfilling φ(x) = φ(y) contains an internal vertex z with φ(z) > φ(x). On-line ranking number $χ*_r(G)$ of a graph G is a minimum k such that G has a k-ranking constructed step by step if vertices of G are coming and coloured one by one in an arbitrary order; when colouring a vertex, only edges between already present vertices are known. Schiermeyer, Tuza and Voigt proved that $χ*_r(Pₙ) < 3log₂n$ for n ≥ 2. Here we show that $χ*_r(Pₙ) ≤ 2⎣log₂n⎦+1$. The same upper bound is obtained for $χ*_r(Cₙ)$,n ≥ 3.
4
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On Maximum Weight of a Bipartite Graph of Given Order and Size

81%
Horňák M., Jendrol’ S., Schiermeyer I.
Discussiones Mathematicae Graph Theory
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2013
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tom 33
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nr 1
147-165
EN
The weight of an edge xy of a graph is defined to be the sum of degrees of the vertices x and y. The weight of a graph G is the minimum of weights of edges of G. More than twenty years ago Erd˝os was interested in finding the maximum weight of a graph with n vertices and m edges. This paper presents a complete solution of a modification of the above problem in which a graph is required to be bipartite. It is shown that there is a function w*(n,m) such that the optimum weight is either w*(n,m) or w*(n,m) + 1.
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