A Brooks Type Theorem for the Maximum Local Edge Connectivity

A Brooks Type Theorem for the Maximum Local Edge Connectivity

For a graph G, let χ(G) and λ(G) denote the chromatic number of G and the maximum local edge connectivity of G, respectively. A result of Dirac implies that every graph G satisfies χ(G) 6 λ(G) + 1. In this paper we characterize the graphs G for which χ(G) = λ(G) + 1. The case λ(G) = 3 was already solved by Aboulker, Brettell, Havet, Marx, and Trotignon. We show that a graph G with λ(G) = k > 4 ...

A Brooks-type Theorem for the Bandwidth of Interval Graphs

Let G be an interval graph. The layout that arranges the intervals in order by right endpoint easily shows that the bandwidth of G is at most its maximum degree ∆. Hence, if G contains a clique of size ∆ + 1, then its bandwidth must be ∆. In this paper we prove a Brooks-type bound on the bandwidth of interval graphs. Namely, the bandwidth of an interval graph is at most ∆, with equality if and ...

A Brooks-Type Theorem for the Generalized List T-Coloring

A note on a Brooks’ type theorem for DP-coloring

Dvořák and Postle [8] introduced aDP-coloring of a simple graph as a generalization of a list-coloring. They proved a Brooks’ type theorem for a DP-coloring, and Bernsheteyn, Kostochka and Pron [5] extended it to a DP-coloring of multigraphs. However, detailed structure when a multigraph does not admit a DP-coloring was not specified in [5]. In this note, we make this point clear and give the c...

About a Brooks-type theorem for improper colouring

A graph is k-improperly -colourable if its vertices can be partitioned into parts such that each part induces a subgraph of maximum degree at most k. A result of Lovász states that for any graph G, such a partition exists if ≥ ⌈ Δ(G)+1 k+1 ⌉ . When k = 0, this bound can be reduced by ∗ This work was partially supported by the Égide eco-net project 16305SB. † Partially supported by the Brazilian...