106,283 research outputs found
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
A sufficient condition guaranteeing large cycles in graphs
AbstractWe generalize Bedrossian-Chen-Schelp's condition (1993) for the existence of large cycles in graphs, and give infinitely many examples of graphs which fulfill the new condition for hamiltonicity, while the related condition by Bedrossian, Chen, and Schelp is not fulfilled
A bandwidth theorem for approximate decompositions
We provide a degree condition on a regular -vertex graph which ensures
the existence of a near optimal packing of any family of bounded
degree -vertex -chromatic separable graphs into . In general, this
degree condition is best possible.
Here a graph is separable if it has a sublinear separator whose removal
results in a set of components of sublinear size. Equivalently, the
separability condition can be replaced by that of having small bandwidth. Thus
our result can be viewed as a version of the bandwidth theorem of B\"ottcher,
Schacht and Taraz in the setting of approximate decompositions.
More precisely, let be the infimum over all
ensuring an approximate -decomposition of any sufficiently large regular
-vertex graph of degree at least . Now suppose that is an
-vertex graph which is close to -regular for some and suppose that is a sequence of bounded
degree -vertex -chromatic separable graphs with . We show that there is an edge-disjoint packing of
into .
If the are bipartite, then is sufficient. In
particular, this yields an approximate version of the tree packing conjecture
in the setting of regular host graphs of high degree. Similarly, our result
implies approximate versions of the Oberwolfach problem, the Alspach problem
and the existence of resolvable designs in the setting of regular host graphs
of high degree.Comment: Final version, to appear in the Proceedings of the London
Mathematical Societ
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