Maximum edge-cuts in cubic graphs with large girth and in random cubic graphs

We show that for every cubic graph G with sufficiently large girth there exists a probability distribution on edge-cuts of G such that each edge is in a randomly chosen cut with probability at least 0.88672. This implies that G contains an edge-cut of size at least 1.33008n, where n is the number of vertices of G, and has fractional cut covering number at most 1.127752. The lower bound on the size of maximum edge-cut also applies to random cubic graphs. Specifically, a random n-vertex cubic graph a.a.s. contains an edge cut of size 1.33008n

Fractional colorings of cubic graphs with large girth

We show that every (sub)cubic n-vertex graph with sufficiently large girth has fractional chromatic number at most 2.2978 which implies that it contains an independent set of size at least 0.4352n. Our bound on the independence number is valid to random cubic graphs as well as it improves existing lower bounds on the maximum cut in cubic graphs with large girth

Fractional colorings of cubic graphs with large girth

International audienceWe show that every (sub)cubic n-vertex graph with sufficiently large girth has fractional chromatic number at most 2.2978, which implies that it contains an independent set of size at least 0.4352n. Our bound on the independence number is valid for random cubic graphs as well, as it improves existing lower bounds on the maximum cut in cubic graphs with large girth

Vlastnosti grafů velkého obvodu

V práci zkoumáme dva náhodné procesy pro kubické grafy velkého obvodu. První proces nalezne pravděpodobnostní distribuci na hranových řezech takovou, že každá hrana je v náhodně vybraném řezu s pravděpodobností alespoň 0.88672. Jako důsledek odvodíme dolní odhad na velikost největšího řezu pro kubické grafy velkého obvodu a pro náhodné kubické grafy, a dále též horní odhad na váhu nejmenšího zlomkového pokrytí hranovými řezy pro kubické grafy velkého obvodu. Druhý proces nalezne pravděpodobnostní distribuci na nezavislých množinách takovou, že každý vrchol je v nezávislé množině s pravděpodobností alespoň 0.4352. Z toho plyne dolní odhad na velikost největší nezavíslé množiny pro kubické grafy velkého obvodu a pro náhodné kubické grafy, a dále též horní odhad na zlomkovou barevnost pro kubické grafy velkého obvodu.In this work we study two random procedures in cubic graphs with large girth. The first procedure finds a probability distribution on edge-cuts such that each edge is in a randomly chosen cut with probability at least 0.88672. As corollaries, we derive lower bounds for the size of maximum cut in cubic graphs with large girth and in random cubic graphs, and also an upper bound for the fractional cut covering number in cubic graphs with large girth. The second procedure finds a probability distribution on independent sets such that each vertex is in an independent set with probability at least 0.4352. This implies lower bounds for the size of maximum independent set in cubic graphs with large girth and in random cubic graphs, as well as an upper bound for the fractional chromatic number in cubic graphs with large girth.Department of Applied MathematicsKatedra aplikované matematikyFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult