Extremal clique coverings of complementary graphs

Clique coverings and claw-free graphs

“…This also implies the upper bound on scp(G) · scp(G). These upper bounds improve the results of [7].…”

Regarding Two Conjectures on Clique and Biclique Partitions

“…In 1986, de Caen, Erdős, Pullman and Wormald [4] investigated the maximum value of cp(G) + cp(G) over the set G n of all graphs G on n vertices, and proved that…”

“…They hypothesized that the lower bound 7n 2 25 is nearly tight, and left closing the gap in (1.3) as an open problem. 1 Problem 3 (de Caen et al [4]). Find c such that…”

“…In Section 3.1, we show that the family of graphs constructed in [4] can actually be modified to improve the lower bound in (1.3). Specifically, we show that the constant c in (3) must be at least 23 82 > 7 25 .…”

“…The construction in our proof of Theorem 4 is based on the original construction in [4], and the calculation of its clique partition number makes use of certain facts shown in [4] and [17]. Here we include the entire argument for the reader's convenience.…”