64,709 research outputs found
On Structure of Some Plane Graphs with Application to Choosability
AbstractA graph G=(V, E) is (x, y)-choosable for integers x>y⩾1 if for any given family {A(v)∣v∈V} of sets A(v) of cardinality x, there exists a collection {B(v)∣v∈V} of subsets B(v)⊂A(v) of cardinality y such that B(u)∩B(v)=∅ whenever uv∈E(G). In this paper, structures of some plane graphs, including plane graphs with minimum degree 4, are studied. Using these results, we may show that if G is free of k-cycles for some k∈{3, 4, 5, 6}, or if any two triangles in G have distance at least 2, then G is (4m, m)-choosable for all nonnegative integers m. When m=1, (4m, m)-choosable is simply 4-choosable. So these conditions are also sufficient for a plane graph to be 4-choosable
An improvement of sufficient condition for -leaf-connected graphs
For integer a graph is called -leaf-connected if and given any subset with always has a
spanning tree such that is precisely the set of leaves of Thus a
graph is -leaf-connected if and only if it is Hamilton-connected. In this
paper, we present a best possible condition based upon the size to guarantee a
graph to be -leaf-connected, which not only improves the results of Gurgel
and Wakabayashi [On -leaf-connected graphs, J. Combin. Theory Ser. B 41
(1986) 1-16] and Ao, Liu, Yuan and Li [Improved sufficient conditions for
-leaf-connected graphs, Discrete Appl. Math. 314 (2022) 17-30], but also
extends the result of Xu, Zhai and Wang [An improvement of spectral conditions
for Hamilton-connected graphs, Linear Multilinear Algebra, 2021]. Our key
approach is showing that an -closed non--leaf-connected graph must
contain a large clique if its size is large enough. As applications, sufficient
conditions for a graph to be -leaf-connected in terms of the (signless
Laplacian) spectral radius of or its complement are also presented.Comment: 15 pages, 2 figure
Shilla distance-regular graphs
A Shilla distance-regular graph G (say with valency k) is a distance-regular
graph with diameter 3 such that its second largest eigenvalue equals to a3. We
will show that a3 divides k for a Shilla distance-regular graph G, and for G we
define b=b(G):=k/a3. In this paper we will show that there are finitely many
Shilla distance-regular graphs G with fixed b(G)>=2. Also, we will classify
Shilla distance-regular graphs with b(G)=2 and b(G)=3. Furthermore, we will
give a new existence condition for distance-regular graphs, in general.Comment: 14 page
- …