The Harary-Hill conjecture states that the minimum number of crossings in a drawing of the complete graph Kn is Z(n) := 1 4 ⌊ n 2 ⌋ ⌊ n−1 2 ⌋ ⌊ n−2 2 ⌋ ⌊ n−3 2 ⌋ . This conjecture was recently proved for 2-page book drawings of Kn. As an extension of this technique, we prove the conjecture for monotone drawings of Kn, that is, drawings where all vertices have different x-coordinates and the edg...

In [5], Reed conjectures that every graph satisfies χ ≤ ⌈ω+∆+1 2 ⌉ . We prove this holds for graphs with disconnected complement. Combining this fact with a result of Molloy proves the conjecture for graphs satisfying χ > ⌈ n 2 ⌉ . Generalizing this we prove that the conjecture holds for graphs satisfying χ > n+3−α 2 . It follows that the conjecture holds for graphs satisfying ∆ ≥ n + 2 − (α +√...

A locally irregular multigraph is a whose adjacent vertices have distinct degrees. The edge coloring an of G such that every color induces submultigraph G. We say colorable if it admits and we denote by lir(G) the chromatic index G, which smallest number colors required in conjecture for connected graph not isomorphic to K2, 2G obtained from doubling each lir(2G)≤2. This concept closely related...