The Threshold for the Erdős, Jacobson and Lehel Conjecture to Be True

Abstract

Let σ(k, n) be the smallest even integer such that each n–term positive graphic sequence with term sum at least σ(k, n) can be realized by a graph containing a clique of k + 1 vertices. Erdős et al. (Graph Theory, 1991, 439–449) conjectured that σ(k, n) = (k −1)(2n−k)+2. Li et al. (Science in China, 1998, 510–520) proved that the conjecture is true for k ≥ 5 and $n \geqslant {\left( {\begin{array}{*{20}c} {k} \\ {2} \\ \end{array} } \right)} + 3$, and raised the problem of determining the smallest integer N(k) such that the conjecture holds for n ≥ N(k). They also determined the values of N(k) for 2 ≤ k ≤ 7, and proved that ${\left\lceil {\frac{{5k - 1}} {2}} \right\rceil } \leqslant N{\left( k \right)} \leqslant {\left( {\begin{array}{*{20}c} {k} \\ {2} \\ \end{array} } \right)} + 3$ for k ≥ 8. In this paper, we determine the exact values of σ(k, n) for n ≥ 2k+3 and k ≥ 6. Therefore, the problem of determining σ(k, n) is completely solved. In addition, we prove as a corollary that $N{\left( k \right)} = {\left\lceil {\frac{{5k - 1}} {2}} \right\rceil }$ for k ≥ 6.