Jessica McDonald

A minimum degree condition forcing complete graph immersion

An immersion of a graph H into a graph G is a one-to-one mapping f: V (H) → V (G) and a collection of edge-disjoint paths in G, one for each edge of H, such that the path Puv corresponding to edge uv has endpoints f(u) and f(v). The immersion is strong if the paths Puv are internally disjoint from f(V (H)). It is proved that for every positive integer Ht, every simple graph of minimum degree at least 200t contains a strong immersion of the complete graph Kt. For dense graphs one can say even more. If the graph has order n and has 2cn2 edges, then there is a strong immersion of the complete graph on at least c2n vertices in G in which each path Puv is of length 2. As an application of these results, we resolve a problem raised by Paul Seymour by proving that the line graph of every simple graph with average degree d has a clique minor of order at least cd3/2, where c>0 is an absolute constant.For small values of t, 1≤t≤7, every simple graph of minimum degree at least t−1 contains an immersion of Kt (Lescure and Meyniel [13], DeVos et al. [6]). We provide a general class of examples showing that this does not hold when t is large.

Packing Triangles in Weighted Graphs

Tuza conjectured that for every graph

Kempe Equivalence of Edge‐Colorings in Subcubic and Subquartic Graphs

G

Average Degree in Graph Powers

the maximum size

Immersing complete digraphs

\nu

On characterizing Vizing's edge colouring bound

of a set of edge-disjoint triangles and minimum size

Packing Steiner trees

\tau

List-edge-colouring planar graphs with precoloured edges

of a set of edges meeting all triangles satisfy

The list chromatic index of simple graphs whose odd cycles intersect in at most one edge

\tau \leq 2\nu

On a limit of the method of Tashkinov trees for edge-colouring

. We consider an edge-weighted version of this conjecture, which amounts to packing and covering triangles in multigraphs. Several known results about the original problem are shown to be true in this context, and some are improved. In particular, we answer a question of Krivelevich, who proved that