Vol. 10, No. 4, 2017

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The $H$-linked degree-sum parameter for special graph families

Lydia East Kenney and Jeffrey Scott Powell

Vol. 10 (2017), No. 4, 707–720

For a fixed graph H, a graph G is H-linked if any injection f : V (H) V (G) can be extended to an H-subdivision in G. The concept of H-linked generalizes several well-known graph theory concepts such as k-connected, k-linked, and k-ordered. In 2012, Ferrara et al. proved a sharp σ2 (or degree-sum) bound for a graph to be H-linked. In particular, they proved that any graph G with n > 20|E(H)| vertices and σ2(G) n + a(H) 2 is H-linked, where a(H) is a parameter maximized over certain partitions of V (H). However, they do not discuss the calculation of a(H) in their work. In this paper, we prove the exact value of a(H) in the cases when H 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.

H-linked, path, cycle, degree-sum, Ore condition
Mathematical Subject Classification 2010
Primary: 05C35, 05C38
Secondary: 05C83
Received: 5 May 2016
Revised: 1 July 2016
Accepted: 11 July 2016
Published: 7 March 2017

Communicated by Jerrold Griggs
Lydia East Kenney
Smiths Station High School
4228 Lee Rd. 430
Smiths Station, AL 36877
United States
Jeffrey Scott Powell
Department of Mathematics and Computer Science
Samford University
800 Lakeshore Drive
Birmingham, AL 35229
United States