Vizing's 2-Factor Conjecture Involving Large Maximum Degree

Vizing's 2-Factor Conjecture Involving Large Maximum Degree

Let G be a simple graph of order n, and let ∆(G) and χ′(G) denote the maximum degree and chromatic index of G, respectively. Vizing proved that χ′(G) = ∆(G) or ∆(G) + 1. Following this result, G is called edge-chromatic critical if χ′(G) = ∆(G) + 1 and χ′(G − e) = ∆(G) for every e ∈ E(G). In 1968, Vizing conjectured that if G is edge-chromatic critical, then the independence number α(G) ≤ n/2, ...

Gallai's path decomposition conjecture for graphs of small maximum degree

Gallai’s path decomposition conjecture states that the edges of any connected graph on n vertices can be decomposed into at most n+1 2 paths. We confirm that conjecture for all graphs with maximum degree at most five.

Large induced subgraphs with equated maximum degree

For a graph G, denote by fk(G) the smallest number of vertices that must be deleted from G so that the remaining induced subgraph has its maximum degree shared by at least k vertices. It is not difficult to prove that there are graphs for which already f2(G) ≥ √ n(1− o(1)), where n is the number of vertices of G. It is conjectured that fk(G) = Θ( √ n) for every fixed k. We prove this for k = 2,...

Disjoint subgraphs of large maximum degree

Crossing-critical graphs with large maximum degree

A conjecture of Richter and Salazar about graphs that are critical for a fixed crossing number k is that they have bounded bandwidth. A weaker well-known conjecture is that their maximum degree is bounded in terms of k. In this note we disprove these conjectures for every k ≥ 171, by providing examples of k-crossing-critical graphs with arbitrarily large maximum degree. A graph is k-crossing-cr...