### A theorem on nonexistence of a certain type of nearly regular cell-decompositions of the sphere

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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.

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.

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 $\chi {*}_{r}\left(G\right)$ 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 $\chi {*}_{r}\left(P\u2099\right)<3log\u2082n$ for n ≥ 2. Here we show...

The achromatic number of a graph $G$ is the maximum number of colours in a proper vertex colouring of $G$ such that for any two distinct colours there is an edge of $G$ incident with vertices of those two colours. We determine the achromatic number of the Cartesian product of ${K}_{5}$ and ${K}_{n}$ for all $n\le 24$.

We prove that any complete bipartite graph ${K}_{a,b}$, where $a,b$ are even integers, can be decomposed into closed trails with prescribed even lengths.

Let $\mathrm{Lct}\left(G\right)$ denote the set of all lengths of closed trails that exist in an even graph $G$. A sequence $({t}_{1},\cdots ,{t}_{p})$ of elements of $\mathrm{Lct}\left(G\right)$ adding up to $\left|E\right(G\left)\right|$ is $G$-realisable provided there is a sequence $({T}_{1},\cdots ,{T}_{p})$ of pairwise edge-disjoint closed trails in $G$ such that ${T}_{i}$ is of length ${t}_{i}$ for $i=1,\cdots ,p$. The graph $G$ is arbitrarily decomposable into closed trails if all possible sequences are $G$-realisable. In the paper it is proved that if $a\ge 1$ is an odd integer and ${M}_{a,a}$ is a perfect matching in ${K}_{a,a}$, then the graph ${K}_{a,a}-{M}_{a,a}$ is arbitrarily decomposable into closed...

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...

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