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1
Content available remote

Light Graphs In Planar Graphs Of Large Girth

100%
Hudák P., Maceková M., Madaras T., Široczki P.
Discussiones Mathematicae Graph Theory
|
2016
|
tom 36
|
nr 1
227-238
EN
A graph H is defined to be light in a graph family 𝒢 if there exist finite numbers φ(H, 𝒢) and w(H, 𝒢) such that each G ∈ 𝒢 which contains H as a subgraph, also contains its isomorphic copy K with ΔG(K) ≤ φ(H, 𝒢) and ∑x∈V(K) degG(x) ≤ w(H, 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 φ and w for light K1,3 and C10.
2
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The Thickness of Amalgamations and Cartesian Product of Graphs

100%
Yang Y., Chen Y.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 3
561-572
EN
The thickness of a graph is the minimum number of planar spanning subgraphs into which the graph can be decomposed. It is a measurement of the closeness to the planarity of a graph, and it also has important applications to VLSI design, but it has been known for only few graphs. We obtain the thickness of vertex-amalgamation and bar-amalgamation of graphs, the lower and upper bounds for the thickness of edge-amalgamation and 2-vertex-amalgamation of graphs, respectively. We also study the thickness of Cartesian product of graphs, and by using operations on graphs, we derive the thickness of the Cartesian product Kn □ Pm for most values of m and n.
3
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The structure of plane graphs with independent crossings and its applications to coloring problems

81%
Zhang X., Liu G.
Open Mathematics
|
2013
|
tom 11
|
nr 2
308-321
EN
If a graph G has a drawing in the plane in such a way that every two crossings are independent, then we call G a plane graph with independent crossings or IC-planar graph for short. In this paper, the structure of IC-planar graphs with minimum degree at least two or three is studied. By applying their structural results, we prove that the edge chromatic number of G is Δ if Δ ≥ 8, the list edge (resp. list total) chromatic number of G is Δ (resp. Δ + 1) if Δ ≥ 14 and the linear arboricity of G is ℈Δ/2⌊ if Δ ≥ 17, where G is an IC-planar graph and Δ is the maximum degree of G.
4
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Regularity and Planarity of Token Graphs

81%
Carballosa W., Fabila-Monroy R., Leaños J., Rivera L. M.
Discussiones Mathematicae Graph Theory
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2017
|
tom 37
|
nr 3
573-586
EN
Let G = (V, E) be a graph of order n and let 1 ≤ k < n be an integer. The k-token graph of G is the graph whose vertices are all the k-subsets of V, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. In this paper we characterize precisely, for each value of k, which graphs have a regular k-token graph and which connected graphs have a planar k-token graph.
5
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On (p, 1)-total labelling of 1-planar graphs

81%
Zhang X., Yu Y., Liu G.
Open Mathematics
|
2011
|
tom 9
|
nr 6
1424-1434
EN
A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that the (p, 1)-total labelling number of every 1-planar graph G is at most Δ(G) + 2p − 2 provided that Δ(G) ≥ 8p+4 or Δ(G) ≥ 6p+2 and g(G) ≥ 4. As a consequence, the well-known (p, 1)-total labelling conjecture has been confirmed for some 1-planar graphs.
6
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On An Extremal Problem In The Class Of Bipartite 1-Planar Graphs

62%
Czap J., Przybyło J., Škrabuľáková E.
Discussiones Mathematicae Graph Theory
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2016
|
tom 36
|
nr 1
141-151
EN
A graph G = (V, E) is called 1-planar if it admits a drawing in the plane such that each edge is crossed at most once. In this paper, we study bipartite 1-planar graphs with prescribed numbers of vertices in partite sets. Bipartite 1-planar graphs are known to have at most 3n − 8 edges, where n denotes the order of a graph. We show that maximal-size bipartite 1-planar graphs which are almost balanced have not significantly fewer edges than indicated by this upper bound, while the same is not true for unbalanced ones. We prove that the maximal possible size of bipartite 1-planar graphs whose one partite set is much smaller than the other one tends towards 2n rather than 3n. In particular, we prove that if the size of the smaller partite set is sublinear in n, then |E| = (2 + o(1))n, while the same is not true otherwise.
7
Content available remote

Split Euler Tours In 4-Regular Planar Graphs

62%
Couch P., Daniel B. D., Guidry R., Paul Wright W.
Discussiones Mathematicae Graph Theory
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2016
|
tom 36
|
nr 1
23-30
EN
The construction of a homing tour is known to be NP-complete. On the other hand, the Euler formula puts su cient restrictions on plane graphs that one should be able to assert the existence of such tours in some cases; in particular we focus on split Euler tours (SETs) in 3-connected, 4-regular, planar graphs (tfps). An Euler tour S in a graph G is a SET if there is a vertex v (called a half vertex of S) such that the longest portion of the tour between successive visits to v is exactly half the number of edges of G. Among other results, we establish that every tfp G having a SET S in which every vertex of G is a half vertex of S can be transformed to another tfp G′ having a SET S′ in which every vertex of G′ is a half vertex of S′ and G′ has at most one point having a face configuration of a particular class. The various results rely heavily on the structure of such graphs as determined by the Euler formula and on the construction of tfps from the octahedron. We also construct a 2-connected 4-regular planar graph that does not have a SET.
8
Content available remote

Sharp Upper Bounds on the Clar Number of Fullerene Graphs

52%
Gao Y., Zhang H.
Discussiones Mathematicae Graph Theory
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2017
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tom 38
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nr 1
155-163
EN
The Clar number of a fullerene graph with n vertices is bounded above by ⌊n/6⌋ − 2 and this bound has been improved to ⌊n/6⌋ − 3 when n is congruent to 2 modulo 6. We can construct at least one fullerene graph attaining the upper bounds for every even number of vertices n ≥ 20 except n = 22 and n = 30.
9
Content available remote

Gromov hyperbolicity of planar graphs

52%
Cantón A., Granados A., Pestana D., Rodríguez J.
Open Mathematics
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2013
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tom 11
|
nr 10
1817-1830
EN
We prove that under appropriate assumptions adding or removing an infinite amount of edges to a given planar graph preserves its non-hyperbolicity, a result which is shown to be false in general. In particular, we make a conjecture that every tessellation graph of ℝ2 with convex tiles is non-hyperbolic; it is shown that in order to prove this conjecture it suffices to consider tessellation graphs of ℝ2 such that every tile is a triangle and a partial answer to this question is given. A weaker version of this conjecture stating that every tessellation graph of ℝ2 with rectangular tiles is non-hyperbolic is given and partially answered. If this conjecture were true, many tessellation graphs of ℝ2 with tiles which are parallelograms would be non-hyperbolic.
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