2 research outputs found
Fractional refinements of integral theorems
The focus of this thesis is to take theorems which deal with ``integral" objects in graph theory and consider fractional refinements of them to gain additional structure.
A classic theorem of Hakimi says that for an integer , a graph has maximum average degree at most if and only if the graph decomposes into pseudoforests. To find a fractional refinement of this theorem, one simply needs to consider the instances where the maximum average degree is fractional.
We prove that for any positive integers and , if has maximum average degree at most , then decomposes into pseudoforests, where one of pseudoforests has every connected component containing at most edges, and further this pseudoforest is acyclic. The maximum average degree bound is best possible for every choice of and .
Similar to Hakimi's Theorem, a classical theorem of Nash-Williams says that a graph has fractional arborcity at most if and only if decomposes into forests. The Nine Dragon Tree Theorem, proven by Jiang and Yang, provides a fractional refinement of Nash-Williams Theorem. It says, for any positive integers and , if a graph has fractional arboricity at most , then decomposes into forests, where one of the forests has maximum degree .
We prove a strengthening of the Nine Dragon Tree Theorem in certain cases. Let and . Every graph with fractional arboricity at most decomposes into two forests and where has maximum degree , every component of contains at most one vertex of degree , and if , then every component of contains at most edges such that both and .
In fact, when and , we prove that every graph with fractional arboricity decomposes into two forests such that has maximum degree , every component of has at most one vertex of degree , further if a component of has a vertex of degree then it has at most edges, and otherwise a component of has at most edges.
Shifting focus to problems which partition the vertex set, circular colouring provides a way to fractionally refine colouring problems. A classic theorem of Tuza says that every graph with no cycles of length is -colourable. Generalizing this to circular colouring, we get the following:
Let and be relatively prime, with , and let be the element of such that . Let be an edge in a graph . If is -circular-colorable and is not, then lies in at least one
cycle in of length congruent to for some in
. If this does not occur with , then lies in at least two cycles of length and contains a cycle of length .
This theorem is best possible with regards to the number of congruence classes when .
A classic theorem of Gr\"{o}tzsch says that triangle free planar graphs are -colourable. There are many generalizations of this result, however fitting the theme of fractional refinements, Jaeger conjectured that every planar graph of girth admits a homomorphism to . While we make no progress on this conjecture directly, one way to approach the conjecture is to prove critical graphs have large average degree. On this front, we prove:
Every -critical graph which does not have a -colouring and is not or satisfies , and
every triangle free -critical graph satisfies .
In the case of the second theorem, a result of Davies shows there exists infinitely many triangle free -critical graphs satisfying , and hence the second theorem is close to being tight. It also generalizes results of Thomas and Walls, and also Thomassen, that girth graphs embeddable on the torus, projective plane, or Klein bottle are -colourable.
Lastly, a theorem of Cereceda, Johnson, and van den Heuvel, says that given a -connected bipartite planar graph with no separating four-cycles and a -colouring , then one can obtain all -colourings from by changing one vertices' colour at a time if and only if has at most one face of size .
We give the natural generalization of this to circular colourings when