Edge clique covers in graphs with independence number two

Abstract

The edge clique cover number ecc ( G ) of a graph G is the size of the smallest collection of complete subgraphs whose union covers all edges of G. Chen, Jacobson, Kézdy, Lehel, Scheinerman, and Wang conjectured in 2000 that if G is claw‐free, then ecc ( G ) is bounded above by its order (denoted n). Recently, Javadi and Hajebi verified this conjecture for claw‐free graphs with an independence number at least three. We study the edge clique cover number of graphs with independence number two, which are necessarily claw‐free. We give the first known proof of a linear bound in n for ecc ( G ) for such graphs, improving upon the bou nd of O ( n 4 ∕ 3 log 1 ∕ 3 n ) due to Javadi, Maleki, and Omoomi. More precisely we prove that ecc ( G ) is at most the minimum of n + δ ( G ) and 2 n − normalΩ ( n log n ), where δ ( G ) is the minimum degree of G. In the fractional version of the problem, we improve these upper bounds to 3 2 n. We also verify the conjecture for some specific subfamilies, for example, when the edge packing number with respect to cliques (a lower bound for ecc ( G )) equals n, and when G contains no induced subgraph isomorphic to H where H is any fixed graph of order 4.

Edge Clique Covers in Graphs with Independence Number Two: a Special Case

“…Special classes of graphs satisfying this conjecture have been found. For example graphs with diameter three, or 2K 2 -free and co-claw free graphs [1]. Our first result adds one more class of graphs that satisfy the conjecture.…”

“…Indeed, for each edge xy of G, there is a vertex u that is not adjacent to either of its ends, and so xy is covered by the clique N [u]. We thus have the following lemma, proved in [1]: Lemma 10 ( [1]). Let G be a simple graph such that α(G) = 2.…”

“…It can clearly not be of diameter 1 and if it had diameter 4 or more, it would have α(G) ≥ 3. The diameter 3 case has been solved in [1].…”

“…As this is still out of reach, we can try to find upper bounds for the edge clique cover number. The best known bound on ecc(G) for general graphs with independence number two is the minimum of n + δ(G) and 2n − Ω( √ n log n), where δ(G) is the minimum degree of G [1]. Better bounds has been found for some classes of graphs.…”

“…Better bounds has been found for some classes of graphs. For example, it has been shown in [1] that if the chromatic number χ( Ḡ) of the complement of G is 4 then ecc(G) ≤ 3 2 n + 4. The combination of a large class of graphs G with a good bound on the ecc(G) is mostly elusive at this time.…”