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Decomposing highly edge-connected graphs into homomorphic copies of a fixed tree
The Tree Decomposition Conjecture by Bar\'at and Thomassen states that for
every tree there exists a natural number such that the following
holds: If is a -edge-connected simple graph with size divisible by
the size of , then can be edge-decomposed into subgraphs isomorphic to
. So far this conjecture has only been verified for paths, stars, and a
family of bistars. We prove a weaker version of the Tree Decomposition
Conjecture, where we require the subgraphs in the decomposition to be
isomorphic to graphs that can be obtained from by vertex-identifications.
We call such a subgraph a homomorphic copy of . This implies the Tree
Decomposition Conjecture under the additional constraint that the girth of
is greater than the diameter of . As an application, we verify the Tree
Decomposition Conjecture for all trees of diameter at most 4.Comment: 18 page
Decomposing 8-regular graphs into paths of length 4
A -decomposition of a graph is a set of edge-disjoint copies of in
that cover the edge set of . Graham and H\"aggkvist (1989) conjectured
that any -regular graph admits a -decomposition if is a tree
with edges. Kouider and Lonc (1999) conjectured that, in the special
case where is the path with edges, admits a -decomposition
where every vertex of is the end-vertex of exactly two paths
of , and proved that this statement holds when has girth at
least . In this paper we verify Kouider and Lonc's Conjecture for
paths of length
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