# On graphs double-critical with respect to the colouring number

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1 Institute of Mathematics - Technical University of Ilmenau 2 IMADA - Department of Mathematics and Computer Science Odense 3 Research Clinic on Gambling Disorders, Aarhus University Hospital

Abstract : The colouring number col$G$ of a graph $G$ is the smallest integer $k$ for which there is an ordering of the vertices of $G$ such that when removing the vertices of $G$ in the specified order no vertex of degree more than $k-1$ in the remaining graph is removed at any step. An edge $e$ of a graph $G$ is said to be double-col-critical if the colouring number of $G-Ve$ is at most the colouring number of $G$ minus 2. A connected graph G is said to be double-col-critical if each edge of $G$ is double-col-critical. We characterise the double-col-critical graphs with colouring number at most 5. In addition, we prove that every 4-col-critical non-complete graph has at most half of its edges being double-col-critical, and that the extremal graphs are precisely the odd wheels on at least six vertices. We observe that for any integer $k$ greater than 4 and any positive number $ε$, there is a $k$-col-critical graph with the ratio of double-col-critical edges between $1- ε$ and 1.

Keywords : graph characterizations graph colouring degenerate graphs colouring number double-critical graphs

Author: ** Matthias Kriesell - Anders Pedersen - **

Source: https://hal.archives-ouvertes.fr/