A note on 1-planar graphs with minimum degree 7
نویسندگان
چکیده
It is well-known that 1-planar graphs have minimum degree at most 7, and not hard to see some exactly 7. In this note we show any such graph has least 24 vertices, tight.
منابع مشابه
Light edges in 1-planar graphs with prescribed minimum degree
A graph is called 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. We prove that each 1-planar graph of minimum degree δ ≥ 4 contains an edge with degrees of its endvertices of type (4,≤ 13) or (5,≤ 9) or (6,≤ 8) or (7, 7). We also show that for δ ≥ 5 these bounds are best possible and that the list of edges is minimal (in the sense that, for each...
متن کاملA note on 1-planar graphs
A graph is 1-planar if it can be drawn in the plane such that each of its edges is crossed at most once. We prove a conjecture of Czap and Hudák [6] stating that the edge set of every 1-planar graph can be decomposed into a planar graph and a forest. We also provide simple proofs for the following recent results: (i) an n-vertex graph that admits a 1-planar drawing with straight-line edges has ...
متن کاملA Note on Minimum-Segment Drawings of Planar Graphs
A straight-line drawing of a planar graph G is a planar drawing of G, where each vertex is mapped to a point on the Euclidean plane and each edge is drawn as a straight line segment. A segment in a straight-line drawing is a maximal set of edges that form a straight line segment. A minimum-segment drawing of G is a straightline drawing of G, where the number of segments is the minimum among all...
متن کاملDiameter Two Graphs of Minimum Order with Given Degree Set
The degree set of a graph is the set of its degrees. Kapoor et al. [Degree sets for graphs, Fund. Math. 95 (1977) 189-194] proved that for every set of positive integers, there exists a graph of diameter at most two and radius one with that degree set. Furthermore, the minimum order of such a graph is determined. A graph is 2-self- centered if its radius and diameter are two. In this paper for ...
متن کاملEnumeration of Maximal Planar Graphs with minimum degree Five
In this paper, we give a process to enumerate all maximal planar graphs with minimum degree ve, with a xed number of vertices. These graphs are called MPG5 [6]. We have used the algorithm [3] to generate these graphs. For the MPG5 enumeration we have used the general method due to Brenda McKay in [5]; i.e. by computing the orbits set of automorphism group and canonical forms.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Discrete Applied Mathematics
سال: 2021
ISSN: ['1872-6771', '0166-218X']
DOI: https://doi.org/10.1016/j.dam.2020.10.004