A Potential Degree Sequence Problem of the Loebl-Komlós-Sós Conjecture

Abstract

A non-increasing sequence π = (d₁, ⋯, d_n) of nonnegative integers is said to be a graphic sequence if it is realizable by a simple graph G on n vertices. In this case, G is referred to as a realization of π. In terms of graphic sequences, the Loebl-Komlós-Sós conjecture states that for any integers k and n, if π = (d₁, ⋯, d_n) is a graphic sequence with \({d_{\left\lceil {{n \over 2}} \right\rceil}} \ge k\), then every realization of π contains all trees with k edges as subgraphs. This problem can be viewed as a forcible degree sequence problem. In this paper, we consider a potential degree sequence problem of the Loebl-Komlós-Sós conjecture, that is, we prove that for any integers k and n, if π = {d₁, ⋯, d_n) is a graphic sequence with \({d_{\left\lceil {{n \over 2}} \right\rceil}} \ge k\), then there is a realization of π containing all trees with k edges as subgraphs.