62 research outputs found
A list version of graph packing
We consider the following generalization of graph packing. Let and be graphs of order and a bipartite graph. A bijection from
onto is a list packing of the triple if implies and for all . We extend the classical results of Sauer and Spencer and Bollob\'{a}s
and Eldridge on packing of graphs with small sizes or maximum degrees to the
setting of list packing. In particular, we extend the well-known
Bollob\'{a}s--Eldridge Theorem, proving that if , and , then either packs or is one of 7 possible
exceptions. Hopefully, the concept of list packing will help to solve some
problems on ordinary graph packing, as the concept of list coloring did for
ordinary coloring.Comment: 10 pages, 4 figure
The Tur\'an number of sparse spanning graphs
For a graph , the {\em extremal number} is the maximum number of
edges in a graph of order not containing a subgraph isomorphic to . Let
and denote the minimum degree and maximum degree of
, respectively. We prove that for all sufficiently large, if is any
graph of order with , then . The condition on the maximum degree is tight up to a
constant factor. This generalizes a classical result of Ore for the case
, and resolves, in a strong form, a conjecture of Glebov, Person, and
Weps for the case of graphs. A counter-example to their more general conjecture
concerning the extremal number of bounded degree spanning hypergraphs is also
given
On the bipartite graph packing problem
The graph packing problem is a well-known area in graph theory. We consider a
bipartite version and give almost tight conditions on the packability of two
bipartite sequences
Embedding graphs having Ore-degree at most five
Let and be graphs on vertices, where is sufficiently large.
We prove that if has Ore-degree at most 5 and has minimum degree at
least then Comment: accepted for publication at SIAM J. Disc. Mat
Graph packing with constraints on edges
A graph consists of a set of vertices (nodes) and a set of edges (line connecting vertices). Two graphs pack when they have the same number of vertices and we can put them in the same vertex set without overlapping edges. Studies such as Sauer and Spencer, Bollobas and Eldridge, Kostochka and Yu, have shown sufficient conditions, specifically relations between number of edges in the two graphs, for two graphs to pack, but only a few addressed packing with constraints. Kostochka and Yu proved that if e_1e_2 \u3c (1 - \eps)n^2, then and pack with exceptions. We extend this finding by using the language of list packing introduced by Gyori, Kostochka, McConvey, and Yager, and we show that the triple with e_1e_2 + \frac{n-1}{2}\cdot e_3 \u3c (2 - \eps)\binom{n}{2} pack with well-defined exceptions
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