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1

Brouwer's theorem for a square on the basis of Hex theorem

Barcz Eugeniusz

Silesian Journal of Pure and Applied Mathematics

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2022

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Vol. 12, iss. 1

33--44

EN

This work is a continuation of author's work [1] on fixed points. In this work, Brouwer's theorem is proved on the basis of the Hex theorem. In the proof, the author uses, among other things, the lemma about no draw. Two proofs of this lemma are derived. The second proof is a modification of D. Gale's proof [2] and is based on the concept of a walk on the Hex board.

2

Planar packing of cycles and unicyclic graphs

Gorlich A.

Demonstratio Mathematica

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2009

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Vol. 42, nr 4

673-679

EN

We say that a graph G is packable into a complete graph Kn if there are two edge-disjoint subgraphs of Kn both isomorphic to G. It is equivalent to the existence of a permutation a of a vertex set in G such that if an edge xy belongs to E(G), then a(x)cr(y) does not belong to E(G). In 2002 Garcia et al. have shown that a non-star tree T is planary packable into a complete graph Kn. In this paper we show that for any packable cycle Cn except of the case n = 5 and n=7 there exists a planar packing into Kn. We also generalize this result to certain classes of unicyclic graphs.

3

Tree domatic number in graphs

Chen X. G.

Opuscula Mathematica

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2007

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Vol. 27, no. 1

5-11

EN

A dominating set S in a graph G is a tree dominating set of G if the subgraph induced by S is a tree. The tree domatic number of G is the maximum number of pairwise disjoint tree dominating sets in V(G). First, some exact values of and sharp bounds for the tree domatic number are given. Then, we establish a sharp lower bound for the number of edges in a connected graph of given order and given tree domatic number, and we characterize the extremal graphs. Finally, we show that a tree domatic number of a planar graph is at most 4 and give a characterization of planar graphs with the tree domatic number 3.

4

Matrix representation and the analysis of shapes of roofs

Koźniewski E.

Journal Biuletyn of Polish Society for Geometry and Engineering Graphics

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2005

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Vol. 15

9--19

EN

The paper is the supplement of a series of articles devoted to geometry of roofs. Regular roofs generated by k-connected generalized polygon can treated as geometrical configurations in the form((2V+2(K−2))3, (3V+3(K−2))2) and described by means incidence or adjacence matrices. After all, such represention results from the natural graph-theoretical characterization of roofs described in previous sections. so, a regular roof can be described as An incidence matrix mutually related to vertices ↔edges, and as ad-Jacency matrix mutually related to vertices ↔hipped roof ends (in graph-theoretical interpretation for planar graphs:vertices ↔Regions). In order to built all topological types of roofs every case of the adjacency matrix satisfying the condition (10) Has to be studied. Adjacency matrices are already rare matrices for V=6. Therefore such a combinatorical way should be too complicated to be used here. The way leading through algebraic-geometrical analysis roposed in papers [5,6] seems to be more familiar and simple. In paper [6] the analysis of the existence of topological types Of Roofs only for V=8HAS been made. Here we complete the analysis for the remaining numbers V=3,4,5,6,7 of sides of the base of investigated roofs. Key words: geometry of roofs, planar graph, Euler theorem for roofs, equations of roof, straight skeleton.

PL

Praca stanowi uzupełnienie cyklu artykułów poświęconych geometrii dachów. Dachy regularne, generowane przez k-spójne wielokąty uogólnione mogą być traktowane jako konfiguracje geometryczne postaci ((2v+2(k−2))3, (3v+3(k−2))2) i opisywane za pomocą macierzy incydencji lub adjacencji. Reprezentacja taka wynika z naturalnej, grafowej, charakteryzacji dachów. Ale wówczas, w celu opisania wszystkich topologicznych typów, każdy przypadek macierzy adjacencji, spełniający opisany w pracy warunek, musiałby być rozpatrzony. Ponieważ macierze adjacencji są rzadkie (już dla v=6), ich analiza kombinatoryczna, w przypadkach v=6, 7, 8, wymagałaby rozpatrzenia bardzo dużej liczby przypadków. Pozostaje więc zdecydowanie prostsza droga algebraiczno-geometryczna oparta na własnościach grafów dachów. W artykule przeprowadzono analizę kształtów dachów dla v=3, 4, 5, 6, 7 uzupełniając tym samym treść cyklu pierwszych prac na ten temat.