A Note On Reed's Conjecture

A note on Reed's Conjecture about $ω$, $Δ$ and $χ$ with respect to vertices of high degree

A note on Reed's Conjecture about $\omega$, $\Delta$ and $\chi$ with respect to vertices of high degree

Reed conjectured that for every graph, χ ≤ ∆+ω+1 2 holds, where χ, ω and ∆ denote the chromatic number, clique number and maximum degree of the graph, respectively. We develop an algorithm which takes a hypothetical counterexample as input. The output discloses some hidden structures closely related to high vertex degrees. Consequently, we deduce two graph classes where Reed's Conjecture holds:...

A note on Fouquet-Vanherpe’s question and Fulkerson conjecture

‎The excessive index of a bridgeless cubic graph $G$ is the least integer $k$‎, ‎such that $G$ can be covered by $k$ perfect matchings‎. ‎An equivalent form of Fulkerson conjecture (due to Berge) is that every bridgeless‎ ‎cubic graph has excessive index at most five‎. ‎Clearly‎, ‎Petersen graph is a cyclically 4-edge-connected snark with excessive index at least 5‎, ‎so Fouquet and Vanherpe as...

A local strengthening of Reed's ω, Δ, χ conjecture for quasi-line graphs

Reed’s ω, ∆, χ conjecture proposes that every graph satisfies χ ≤ d 12 (∆ + 1 + ω)e; it is known to hold for all claw-free graphs. In this paper we consider a local strengthening of this conjecture. We prove the local strengthening for line graphs, then note that previous results immediately tell us that the local strengthening holds for all quasi-line graphs. Our proofs lead to polytime algori...

A Local Strengthening of Reed's Omega, Delta, Chi Conjecture for Quasi-line Graphs

Reed’s ω, ∆, χ conjecture proposes that every graph satisfies χ ≤ d 12 (∆ + 1 + ω)e; it is known to hold for all claw-free graphs. In this paper we consider a local strengthening of this conjecture. We prove the local strengthening for line graphs, then note that previous results immediately tell us that the local strengthening holds for all quasi-line graphs. Our proofs lead to polytime algori...