For a fixed graph , a graph is -linked if any injection can be extended to an -subdivision in . The concept of -linked generalizes several well-known graph theory concepts such as -connected, -linked, and -ordered. In 2012, Ferrara et al. proved a sharp (or degree-sum) bound for a graph to be -linked. In particular, they proved that any graph with vertices and is -linked, where is a parameter maximized over certain partitions of . However, they do not discuss the calculation of in their work. In this paper, we prove the exact value of in the cases when is a path, a cycle, a union of stars, a complete graph, and a complete bipartite graph. Several of these results lead to new degree-sum conditions for particular graph classes while others provide alternate proofs of previously known degree-sum conditions.
"The $H$-linked degree-sum parameter for special graph families." Involve 10 (4) 707 - 720, 2017. https://doi.org/10.2140/involve.2017.10.707