Combined degree and connectivity conditions for H-linked graphs

TL;DR

This paper investigates conditions under which a graph G is H-linked, focusing on the interplay between minimum degree, connectivity, and the size of G, providing bounds for specific small multigraphs H.

Contribution

The paper establishes bounds for the minimum degree needed for H-linkedness in k-connected graphs, especially for small multigraphs with up to three edges, improving understanding of graph linkage conditions.

Findings

Derived bounds for all multigraphs with up to three edges

Identified optimal bounds up to small additive or multiplicative constants

Analyzed the relationship between connectivity, degree, and H-linkedness

Abstract

For a given multigraph H, a graph G is H-linked, if |G| \geq |H| and for every injective map {\tau}: V (H) \rightarrow V (G), we can find internally disjoint paths in G, such that every edge from uv in H corresponds to a {\tau} (u) - {\tau} (v) path. To guarantee that a G is H-linked, you need a minimum degree larger than |G|/2. This situation changes, if you know that G has a certain connectivity k. Depending on k, even a minimum degree independent of |G| may suffice. Let {\delta}(k, H, N) be the minimum number, such that every k-connected graph G with |G| = N and {\delta}(G) \geq {\delta}(k, H, N) is H-linked. We study bounds for this quantity. In particular, we find bounds for all multigraphs H with at most three edges, which are optimal up to small additive or multiplicative constants.