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1
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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An inequality concerning edges of minor weight in convex 3-polytopes

100%
Fabrici I., Jendrol' S.
Discussiones Mathematicae Graph Theory
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1996
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tom 16
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nr 1
81-87
EN
Let $e_{ij}$ be the number of edges in a convex 3-polytope joining the vertices of degree i with the vertices of degree j. We prove that for every convex 3-polytope there is $20e_{3,3} + 25e_{3,4} + 16e_{3,5} + 10e_{3,6} + 6[2/3]e_{3,7} + 5e_{3,8} + 2[1/2]e_{3,9} + 2e_{3,10} + 16[2/3]e_{4,4} + 11e_{4,5} + 5e_{4,6} + 1[2/3]e_{4,7} + 5[1/3]e_{5,5} + 2e_{5,6} ≥ 120$; moreover, each coefficient is the best possible. This result brings a final answer to the conjecture raised by B. Grünbaum in 1973.
2
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Unique-Maximum Coloring Of Plane Graphs

100%
Fabrici I., Göring F.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
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nr 1
95-102
EN
A unique-maximum k-coloring with respect to faces of a plane graph G is a coloring with colors 1, . . . , k so that, for each face of G, the maximum color occurs exactly once on the vertices of α. We prove that any plane graph is unique-maximum 3-colorable and has a proper unique-maximum coloring with 6 colors.
3
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On Vertices Enforcing a Hamiltonian Cycle

81%
Fabrici I., Hexel E., Jendrol’ S.
Discussiones Mathematicae Graph Theory
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2013
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tom 33
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nr 1
71-89
EN
A nonempty vertex set X ⊆ V (G) of a hamiltonian graph G is called an H-force set of G if every X-cycle of G (i.e. a cycle of G containing all vertices of X) is hamiltonian. The H-force number h(G) of a graph G is defined to be the smallest cardinality of an H-force set of G. In the paper the study of this parameter is introduced and its value or a lower bound for outerplanar graphs, planar graphs, k-connected graphs and prisms over graphs is determined.
4
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On Longest Cycles in Essentially 4-Connected Planar Graphs

81%
Fabrici I., Harant J., Jendroľ S.
Discussiones Mathematicae Graph Theory
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2016
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tom 36
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nr 3
565-575
EN
A planar 3-connected graph G is essentially 4-connected if, for any 3-separator S of G, one component of the graph obtained from G by removing S is a single vertex. Jackson and Wormald proved that an essentially 4-connected planar graph on n vertices contains a cycle C such that [...] . For a cubic essentially 4-connected planar graph G, Grünbaum with Malkevitch, and Zhang showed that G has a cycle on at least ¾ n vertices. In the present paper the result of Jackson and Wormald is improved. Moreover, new lower bounds on the length of a longest cycle of G are presented if G is an essentially 4-connected planar graph of maximum degree 4 or G is an essentially 4-connected maximal planar graph.
5
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Paths of low weight in planar graphs

81%
Fabrici I., Harant J., Jendrol' S.
Discussiones Mathematicae Graph Theory
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2008
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tom 28
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nr 1
121-135
EN
The existence of paths of low degree sum of their vertices in planar graphs is investigated. The main results of the paper are: 1. Every 3-connected simple planar graph G that contains a k-path, a path on k vertices, also contains a k-path P such that for its weight (the sum of degrees of its vertices) in G it holds $w_G(P): = ∑_{u∈ V(P)} deg_G(u) ≤ (3/2)k² + 𝓞(k)$ 2. Every plane triangulation T that contains a k-path also contains a k-path P such that for its weight in T it holds $w_T(P): = ∑_{u∈ V(P)} deg_T(u) ≤ k² +13 k$ 3. Let G be a 3-connected simple planar graph of circumference c(G). If c(G) ≥ σ| V(G)| for some constant σ > 0 then for any k, 1 ≤ k ≤ c(G), G contains a k-path P such that $w_G(P) = ∑_{u∈ V(P)} deg_G(u) ≤ (3/σ + 3)k$.
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