Extensions of Fractional Precolorings show Discontinuous Behavior

We study the following problem: given a real number k and integer d, what is the smallest epsilon such that any fractional (k+epsilon)-precoloring of vertices at pairwise distances at least d of a fractionally k-colorable graph can be extended to a fractional (k+epsilon)-coloring of the whole graph? The exact values of epsilon were known for k=2 and k\ge3 and any d. We determine the exact values of epsilon for k \in (2,3) if d=4, and k \in [2.5,3) if d=6, and give upper bounds for k \in (2,3) if d=5,7, and k \in (2,2.5) if d=6. Surprisingly, epsilon viewed as a function of k is discontinuous for all those values of d

Algorithmic Graph Theory

The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions

Graph Theory

Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem

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 $k$, a graph has maximum average degree at most $2k$ if and only if the graph decomposes into $k$ 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 $k$ and $d$, if $G$ has maximum average degree at most $2k + \frac{2d}{k+d+1}$, then $G$ decomposes into $k+1$ pseudoforests, where one of pseudoforests has every connected component containing at most $d$ edges, and further this pseudoforest is acyclic. The maximum average degree bound is best possible for every choice of $k$ and $d$. Similar to Hakimi's Theorem, a classical theorem of Nash-Williams says that a graph has fractional arborcity at most $k$ if and only if $G$ decomposes into $k$ 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 $k$ and $d$, if a graph $G$ has fractional arboricity at most $k + \frac{d}{k+d+1}$, then $G$ decomposes into $k+1$ forests, where one of the forests has maximum degree $d$. We prove a strengthening of the Nine Dragon Tree Theorem in certain cases. Let $k=1$ and $d \in \{3,4\}$. Every graph with fractional arboricity at most $1 + \frac{d}{d+2}$ decomposes into two forests $T$ and $F$ where $F$ has maximum degree $d$, every component of $F$ contains at most one vertex of degree $d$, and if $d= 4$, then every component of $F$ contains at most $8$ edges $e=xy$ such that both $\deg(x) \geq 3$ and $\deg(y) \geq 3$. In fact, when $k = 1$ and $d=3$, we prove that every graph with fractional arboricity $1 + \frac{3}{5}$ decomposes into two forests $T,F$ such that $F$ has maximum degree $3$, every component of $F$ has at most one vertex of degree $3$, further if a component of $F$ has a vertex of degree $3$ then it has at most $14$ edges, and otherwise a component of $F$ has at most $13$ 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 $1 \bmod k$ is $k$-colourable. Generalizing this to circular colouring, we get the following: Let $k$ and $d$ be relatively prime, with $k>2d$, and let $s$ be the element of $\mathbb{Z}_k$ such that $sd \equiv 1\mod k$. Let $xy$ be an edge in a graph $G$. If $G-xy$ is $(k,d)$-circular-colorable and $G$ is not, then $xy$ lies in at least one cycle in $G$ of length congruent to $is \mod k$ for some $i$ in $\{1,\ldots,d\}$. If this does not occur with $i \in\{1,\ldots,d-1\}$, then $xy$ lies in at least two cycles of length $1 \mod k$ and $G-xy$ contains a cycle of length $0 \mod k$. This theorem is best possible with regards to the number of congruence classes when $k = 2d+1$. A classic theorem of Gr\"{o}tzsch says that triangle free planar graphs are $3$-colourable. There are many generalizations of this result, however fitting the theme of fractional refinements, Jaeger conjectured that every planar graph of girth $4k$ admits a homomorphism to $C_{2k+1}$. 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 $4$-critical graph which does not have a $(7,2)$-colouring and is not $K_{4}$ or $W_{5}$ satisfies $e(G) \geq \frac{17v(G)}{10}$, and every triangle free $4$-critical graph satisfies $e(G) \geq \frac{5v(G)+2}{3}$. In the case of the second theorem, a result of Davies shows there exists infinitely many triangle free $4$-critical graphs satisfying $e(G) = \frac{5v(G) +4}{3}$, and hence the second theorem is close to being tight. It also generalizes results of Thomas and Walls, and also Thomassen, that girth $5$ graphs embeddable on the torus, projective plane, or Klein bottle are $3$-colourable. Lastly, a theorem of Cereceda, Johnson, and van den Heuvel, says that given a $2$-connected bipartite planar graph $G$ with no separating four-cycles and a $3$-colouring $f$, then one can obtain all $3$-colourings from $f$ by changing one vertices' colour at a time if and only if $G$ has at most one face of size $6$. We give the natural generalization of this to circular colourings when $\frac{p}{q} < 4$