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Tytuł:
A Note on Barnette’s Conjecture
Autorzy:
Harant, Jochen
Powiązania:
https://bibliotekanauki.pl/articles/30146724.pdf
Data publikacji:
2013-03-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
planar graph
Hamilton cycle
Barnette’s Conjecture
Opis:
Barnette conjectured that each planar, bipartite, cubic, and 3-connected graph is hamiltonian. We prove that this conjecture is equivalent to the statement that there is a constant c > 0 such that each graph G of this class contains a path on at least c|V (G)| vertices.
Źródło:
Discussiones Mathematicae Graph Theory; 2013, 33, 1; 133-137
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Total Coloring of Claw-Free Planar Graphs
Autorzy:
Liang, Zuosong
Powiązania:
https://bibliotekanauki.pl/articles/32304195.pdf
Data publikacji:
2022-08-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total coloring
total coloring conjecture
planar graph
claw
Opis:
A total coloring of a graph is an assignment of colors to both its vertices and edges so that adjacent or incident elements acquire distinct colors. Let Δ(G) be the maximum degree of G. Vizing conjectured that every graph has a total (Δ + 2)-coloring. This Total Coloring Conjecture remains open even for planar graphs, for which the only open case is Δ = 6. Claw-free planar graphs have Δ ≤ 6. In this paper, we prove that the Total Coloring Conjecture holds for claw-free planar graphs.
Źródło:
Discussiones Mathematicae Graph Theory; 2022, 42, 3; 771-777
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Light Graphs In Planar Graphs Of Large Girth
Autorzy:
Hudák, Peter
Maceková, Mária
Madaras, Tomáš
Široczki, Pavol
Powiązania:
https://bibliotekanauki.pl/articles/31341096.pdf
Data publikacji:
2016-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
planar graph
girth
light graph
Opis:
A graph \( H \) is defined to be light in a graph family if there exist finite numbers \( \phi (H, \mathcal{G} ) \) and \( w(H,\mathcal{G} ) \) such that each \( G \in \mathcal{G} \) which contains \( H \) as a subgraph, also contains its isomorphic copy \( K \) with \( \Delta_G (K) \le \phi (H, \mathcal{G} ) \) and \( \Sigma_{ x \in V(K) } \text{ deg}_G (x) \le w(H, \mathcal{G}) \). In this paper, we investigate light graphs in families of plane graphs of minimum degree 2 with prescribed girth and no adjacent 2-vertices, specifying several necessary conditions for their lightness and providing sharp bounds on \( \phi \) and w for light \( K_{1,3} \) and \( C_{10} \).
Źródło:
Discussiones Mathematicae Graph Theory; 2016, 36, 1; 227-238
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Strong Edge-Coloring Of Planar Graphs
Autorzy:
Song, Wen-Yao
Miao, Lian-Ying
Powiązania:
https://bibliotekanauki.pl/articles/31341618.pdf
Data publikacji:
2017-11-27
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
strong edge-coloring
strong chromatic index
planar graph
dis- charging method
Opis:
A strong edge-coloring of a graph is a proper edge-coloring where each color class induces a matching. We denote by $ \chi_s^' (G) $ the strong chromatic index of $G$ which is the smallest integer $k$ such that $G$ can be strongly edge-colored with $k$ colors. It is known that every planar graph $G$ has a strong edge-coloring with at most $ 4 \Delta (G) + 4 $ colors [R.J. Faudree, A. Gyárfás, R.H. Schelp and Zs. Tuza, The strong chromatic index of graphs, Ars Combin. 29B (1990) 205–211]. In this paper, we show that if $G$ is a planar graph with $ g \ge 5$, then $ \chi_s^' (G) \le 4 \Delta (G) − 2 $ when $ \Delta (G) \ge 6 $ and $ \chi_s^' (G) \le 19 $ when $ \Delta (G) = 5 $, where $g$ is the girth of $G$.
Źródło:
Discussiones Mathematicae Graph Theory; 2017, 37, 4; 845-857
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On Longest Cycles in Essentially 4-Connected Planar Graphs
Autorzy:
Fabrici, Igor
Harant, Jochen
Jendroľ, Stanislav
Powiązania:
https://bibliotekanauki.pl/articles/31340878.pdf
Data publikacji:
2016-08-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
planar graph
longest cycle
Opis:
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 $ |V(C)| \ge \frac{2n+4}{5} $. For a cubic essentially 4-connected planar graph $G$, Grünbaum with Malkevitch, and Zhang showed that $G$ has a cycle on at least $ \frac{3}{4} 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.
Źródło:
Discussiones Mathematicae Graph Theory; 2016, 36, 3; 565-575
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Neighbor Product Distinguishing Total Colorings of Planar Graphs with Maximum Degree at least Ten
Autorzy:
Dong, Aijun
Li, Tong
Powiązania:
https://bibliotekanauki.pl/articles/32227944.pdf
Data publikacji:
2021-11-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
total coloring
neighbor product distinguishing coloring
planar graph
Opis:
A proper [k]-total coloring c of a graph G is a proper total coloring c of G using colors of the set [k] = {1, 2, . . ., k}. Let p(u) denote the product of the color on a vertex u and colors on all the edges incident with u. For each edge uv ∈ E(G), if p(u) ≠ p(v), then we say the coloring c distinguishes adjacent vertices by product and call it a neighbor product distinguishing k-total coloring of G. By X″∏(G), we denote the smallest value of k in such a coloring of G. It has been conjectured by Li et al. that Δ(G) + 3 colors enable the existence of a neighbor product distinguishing total coloring. In this paper, by applying the Combinatorial Nullstellensatz, we obtain that the conjecture holds for planar graph with Δ(G) ≥ 10. Moreover, for planar graph G with Δ(G) ≥ 11, it is neighbor product distinguishing (Δ(G) + 2)-total colorable, and the upper bound Δ(G) + 2 is tight.
Źródło:
Discussiones Mathematicae Graph Theory; 2021, 41, 4; 981-999
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Neighbor Sum Distinguishing Total Chromatic Number of Planar Graphs without 5-Cycles
Autorzy:
Zhao, Xue
Xu, Chang-Qing
Powiązania:
https://bibliotekanauki.pl/articles/32083807.pdf
Data publikacji:
2020-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
neighbor sum distinguishing total coloring
discharging method
planar graph
Opis:
For a given graph $ G = (V (G), E(G)) $, a proper total coloring $ \phi : V (G) \cup E(G) $ $ \rightarrow {1, 2, . . ., k} $ is neighbor sum distinguishing if $ f(u) \ne f(v) $ for each edge $ uv \in E(G) $, where $ f(v) = \Sigma_{ uv \in E(G) } $ $ \phi (uv) + \phi (v) $, $ v \in V (G) $. The smallest integer $k$ in such a coloring of $G$ is the neighbor sum distinguishing total chromatic number, denoted by $ \chi_\Sigma^{''} (G) $. Pilśniak and Woźniak first introduced this coloring and conjectured that $ \chi_\Sigma^{''}(G) \le \Delta (G)+3 $ for any graph with maximum degree $ \Delta (G) $. In this paper, by using the discharging method, we prove that for any planar graph $G$ without 5-cycles, $ \chi_\Sigma^{''} (G) \le \text{max} \{ \Delta (G)+2, 10 \} $. The bound $ \Delta (G) + 2 $ is sharp. Furthermore, we get the exact value of $ \chi_\Sigma^{''} (G) $ if $ \Delta (G) \ge 9 $.
Źródło:
Discussiones Mathematicae Graph Theory; 2020, 40, 1; 243-253
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Precise Upper Bound for the Strong Edge Chromatic Number of Sparse Planar Graphs
Autorzy:
Borodin, Oleg V.
Ivanova, Anna O.
Powiązania:
https://bibliotekanauki.pl/articles/30098005.pdf
Data publikacji:
2013-09-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
planar graph
edge coloring
2-distance coloring
strong edgecoloring
Opis:
We prove that every planar graph with maximum degree $ \Delta $ is strong edge $ (2 \Delta − 1)$-colorable if its girth is at least $ 40 [ \frac{\Delta}{2} ] +1 $. The bound $ 2 \Delta −1 $ is reached at any graph that has two adjacent vertices of degree $ \Delta $ .
Źródło:
Discussiones Mathematicae Graph Theory; 2013, 33, 4; 759-770
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
List Edge Coloring of Planar Graphs without 6-Cycles with Two Chords
Autorzy:
Hu, Linna
Sun, Lei
Wu, Jian-Liang
Powiązania:
https://bibliotekanauki.pl/articles/32083824.pdf
Data publikacji:
2021-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
planar graph
edge choosable
list edge chromatic number
chord
Opis:
A graph G is edge-L-colorable if for a given edge assignment L = {L(e) : e ∈ E(G)}, there exists a proper edge-coloring φ of G such that φ(e) ∈ L(e) for all e ∈ E(G). If G is edge-L-colorable for every edge assignment L such that |L(e)| ≥ k for all e ∈ E(G), then G is said to be edge-k-choosable. In this paper, we prove that if G is a planar graph without 6-cycles with two chords, then G is edge-k-choosable, where k = max{7, Δ(G) + 1}, and is edge-t-choosable, where t = max{9, Δ(G)}.
Źródło:
Discussiones Mathematicae Graph Theory; 2021, 41, 1; 199-211
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
On the Hamiltonian Number of a Plane Graph
Autorzy:
Lewis, Thomas M.
Powiązania:
https://bibliotekanauki.pl/articles/31343593.pdf
Data publikacji:
2019-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
Hamiltonian cycle
Hamiltonian walk
Hamiltonian number
Hamiltonian spectrum
Grinberg’s theorem
planar graph
Opis:
The Hamiltonian number of a connected graph is the minimum of the lengths of the closed spanning walks in the graph. In 1968, Grinberg published a necessary condition for the existence of a Hamiltonian cycle in a plane graph, formulated in terms of the degrees of its faces. We show how Grinberg’s theorem can be adapted to provide a lower bound on the Hamiltonian number of a plane graph.
Źródło:
Discussiones Mathematicae Graph Theory; 2019, 39, 1; 171-181
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Incidence Coloring—Cold Cases
Autorzy:
Kardoš, František
Maceková, Mária
Mockovčiaková, Martina
Sopena, Éric
Soták, Roman
Powiązania:
https://bibliotekanauki.pl/articles/32083735.pdf
Data publikacji:
2020-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
incidence coloring
incidence chromatic number
planar graph
maximum average degree
Opis:
An incidence in a graph G is a pair (v, e) where v is a vertex of G and e is an edge of G incident to v. Two incidences (v, e) and (u, f) are adjacent if at least one of the following holds: (i) v = u, (ii) e = f, or (iii) edge vu is from the set {e, f}. An incidence coloring of G is a coloring of its incidences assigning distinct colors to adjacent incidences. The minimum number of colors needed for incidence coloring of a graph is called the incidence chromatic number. It was proved that at most Δ(G) + 5 colors are enough for an incidence coloring of any planar graph G except for Δ(G) = 6, in which case at most 12 colors are needed. It is also known that every planar graph G with girth at least 6 and Δ(G) ≥ 5 has incidence chromatic number at most Δ(G) + 2. In this paper we present some results on graphs regarding their maximum degree and maximum average degree. We improve the bound for planar graphs with Δ(G) = 6. We show that the incidence chromatic number is at most Δ(G) + 2 for any graph G with mad(G) < 3 and Δ(G) = 4, and for any graph with mad(G)<103 and Δ(G) ≥ 8.
Źródło:
Discussiones Mathematicae Graph Theory; 2020, 40, 1; 345-354
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
All Tight Descriptions of 3-Stars in 3-Polytopes with Girth 5
Autorzy:
Borodin, Oleg V.
Ivanova, Anna O.
Powiązania:
https://bibliotekanauki.pl/articles/31342193.pdf
Data publikacji:
2017-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
3-polytope
planar graph
structure properties
k -star
Opis:
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 of degree at most 4 having three 3-neighbors. Furthermore, we show that there are only these two tight descriptions of 3-stars in P5s. Also, we give a tight description of stars with at least three rays in P5s and pose a problem of describing all such descriptions. Finally, we prove a structural theorem about P5s that might be useful in further research.
Źródło:
Discussiones Mathematicae Graph Theory; 2017, 37, 1; 5-12
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
L(2, 1)-Labelings of Some Families of Oriented Planar Graphs
Autorzy:
Sen, Sagnik
Powiązania:
https://bibliotekanauki.pl/articles/30147217.pdf
Data publikacji:
2014-02-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
homomorphism
planar graph
girth
partial k-tree
outerplanar graph
cactus
2-dipath L(2, 1)-labeling
oriented L(2, 1)-labeling
Opis:
In this paper we determine, or give lower and upper bounds on, the 2-dipath and oriented L(2, 1)-span of the family of planar graphs, planar graphs with girth 5, 11, 16, partial k-trees, outerplanar graphs and cacti.
Źródło:
Discussiones Mathematicae Graph Theory; 2014, 34, 1; 31-48
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Describing Neighborhoods of 5-Vertices in 3-Polytopes with Minimum Degree 5 and Without Vertices of Degrees from 7 to 11
Autorzy:
Borodin, Oleg V.
Ivanova, Anna O.
Kazak, Olesya N.
Powiązania:
https://bibliotekanauki.pl/articles/31342287.pdf
Data publikacji:
2018-08-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
planar graph
structure properties
3-polytope
neighborhood
Opis:
In 1940, Lebesgue proved that every 3-polytope contains a 5-vertex for which the set of degrees of its neighbors is majorized by one of the following sequences: (6, 6, 7, 7, 7), (6, 6, 6, 7, 9), (6, 6, 6, 6, 11), (5, 6, 7, 7, 8), (5, 6, 6, 7, 12), (5, 6, 6, 8, 10), (5, 6, 6, 6, 17), (5, 5, 7, 7, 13), (5, 5, 7, 8, 10), (5, 5, 6, 7, 27), (5, 5, 6, 6, ∞), (5, 5, 6, 8, 15), (5, 5, 6, 9, 11), (5, 5, 5, 7, 41), (5, 5, 5, 8, 23), (5, 5, 5, 9, 17), (5, 5, 5, 10, 14), (5, 5, 5, 11, 13). In this paper we prove that every 3-polytope without vertices of degree from 7 to 11 contains a 5-vertex for which the set of degrees of its neighbors is majorized by one of the following sequences: (5, 5, 6, 6, ∞), (5, 6, 6, 6, 15), (6, 6, 6, 6, 6), where all parameters are tight.
Źródło:
Discussiones Mathematicae Graph Theory; 2018, 38, 3; 615-625
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Tytuł:
Light Minor 5-Stars in 3-Polytopes with Minimum Degree 5 and No 6-Vertices
Autorzy:
Borodin, Oleg V.
Ivanova, Anna O.
Vasil’eva, Ekaterina I.
Powiązania:
https://bibliotekanauki.pl/articles/31348169.pdf
Data publikacji:
2020-11-01
Wydawca:
Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
Tematy:
planar map
planar graph
3-polytope
structural properties
5-star
weight
height
Opis:
In 1940, Lebesgue gave an approximate description of the neighborhoods of 5-vertices in the class P5 of 3-polytopes with minimum degree 5. Given a 3-polytope P, by w(P) denote the minimum of the degree-sum (weight) of the neighborhoods of 5-vertices (minor 5-stars) in P. In 1996, Jendrol’ and Madaras showed that if a polytope P in P5 is allowed to have a 5-vertex adjacent to four 5-vertices, then w(P) can be arbitrarily large. For each P in P5 without vertices of degree 6 and 5-vertices adjacent to four 5-vertices, it follows from Lebesgue’s Theorem that w(P) ≤ 68. Recently, this bound was lowered to w(P) ≤ 55 by Borodin, Ivanova, and Jensen and then to w(P) ≤ 51 by Borodin and Ivanova. In this note, we prove that every such polytope P satisfies w(P) ≤ 44, which bound is sharp.
Źródło:
Discussiones Mathematicae Graph Theory; 2020, 40, 4; 985-994
2083-5892
Pojawia się w:
Discussiones Mathematicae Graph Theory
Dostawca treści:
Biblioteka Nauki
Artykuł
Wyświetlanie 1-15 z 19

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