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### An extension of Kotzig’s theorem on the minimum weight of edges in $3$-polytopes

Mathematica Slovaca

### A structural property of planar graphs and the simultaneous colouring of their edges and faces

Mathematica Slovaca

### Precise lower bound for the number of edges of minor weight in planar maps

Mathematica Slovaca

### Precise Upper Bound for the Strong Edge Chromatic Number of Sparse Planar Graphs

Discussiones Mathematicae Graph Theory

We prove that every planar graph with maximum degree ∆ is strong edge (2∆−1)-colorable if its girth is at least 40 [...] +1. The bound 2∆−1 is reached at any graph that has two adjacent vertices of degree ∆.

### Short cycles of low weight in normal plane maps with minimum degree 5

Discussiones Mathematicae Graph Theory

In this note, precise upper bounds are determined for the minimum degree-sum of the vertices of a 4-cycle and a 5-cycle in a plane triangulation with minimum degree 5: w(C₄) ≤ 25 and w(C₅) ≤ 30. These hold because a normal plane map with minimum degree 5 must contain a 4-star with $w\left({K}_{1,4}\right)\le 30$. These results answer a question posed by Kotzig in 1979 and recent questions of Jendrol’ and Madaras.

### 2-distance 4-colorability of planar subcubic graphs with girth at least 22

Discussiones Mathematicae Graph Theory

The trivial lower bound for the 2-distance chromatic number χ₂(G) of any graph G with maximum degree Δ is Δ+1. It is known that χ₂ = Δ+1 if the girth g of G is at least 7 and Δ is large enough. There are graphs with arbitrarily large Δ and g ≤ 6 having χ₂(G) ≥ Δ+2. We prove the 2-distance 4-colorability of planar subcubic graphs with g ≥ 22.

### All Tight Descriptions of 3-Stars in 3-Polytopes with Girth 5

Discussiones Mathematicae Graph Theory

Lebesgue (1940) proved that every 3-polytope P5 of girth 5 has a path of three vertices of degree 3. Madaras (2004) refined this by showing that every P5 has a 3-vertex with two 3-neighbors and the third neighbor of degree at most 4. This description of 3-stars in P5s is tight in the sense that no its parameter can be strengthened due to the dodecahedron combined with the existence of a P5 in which every 3-vertex has a 4-neighbor. We give another tight description of 3-stars in P5s: there is a vertex...

### On the Weight of Minor Faces in Triangle-Free 3-Polytopes

Discussiones Mathematicae Graph Theory

The weight w(f) of a face f in a 3-polytope is the degree-sum of vertices incident with f. It follows from Lebesgue’s results of 1940 that every triangle-free 3-polytope without 4-faces incident with at least three 3-vertices has a 4-face with w ≤ 21 or a 5-face with w ≤ 17. Here, the bound 17 is sharp, but it was still unknown whether 21 is sharp. The purpose of this paper is to improve this 21 to 20, which is best possible.

### 5-Stars of Low Weight in Normal Plane Maps with Minimum Degree 5

Discussiones Mathematicae Graph Theory

It is known that there are normal plane maps M5 with minimum degree 5 such that the minimum degree-sum w(S5) of 5-stars at 5-vertices is arbitrarily large. In 1940, Lebesgue showed that if an M5 has no 4-stars of cyclic type (5, 6, 6, 5) centered at 5-vertices, then w(S5) ≤ 68. We improve this bound of 68 to 55 and give a construction of a (5, 6, 6, 5)-free M5 with w(S5) = 48

### An Extension of Kotzig’s Theorem

Discussiones Mathematicae Graph Theory

In 1955, Kotzig proved that every 3-connected planar graph has an edge with the degree sum of its end vertices at most 13, which is tight. An edge uv is of type (i, j) if d(u) ≤ i and d(v) ≤ j. Borodin (1991) proved that every normal plane map contains an edge of one of the types (3, 10), (4, 7), or (5, 6), which is tight. Cole, Kowalik, and Škrekovski (2007) deduced from this result by Borodin that Kotzig’s bound of 13 is valid for all planar graphs with minimum degree δ at least 2 in which every...

### Decompositions of quadrangle-free planar graphs

Discussiones Mathematicae Graph Theory

W. He et al. showed that a planar graph not containing 4-cycles can be decomposed into a forest and a graph with maximum degree at most 7. This degree restriction was improved to 6 by Borodin et al. We further lower this bound to 5 and show that it cannot be improved to 3.

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