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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.
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...
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...
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...
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