A Brookstype result for sparse critical graphs
Abstract
A graph $G$ is $k${\em critical} if it has chromatic number $k$, but every proper subgraph of $G$ is $(k1)$colorable. Let $f_k(n)$ denote the minimum number of edges in an $n$vertex $k$critical graph. Recently the authors gave a lower bound, $f_k(n) \geq \left\lceil \frac{(k+1)(k2)V(G)k(k3)}{2(k1)}\right\rceil$, that solves a conjecture by Gallai from 1963 and is sharp for every $n\equiv 1\,({\rm mod }\, k1)$. It is also sharp for $k=4$ and every $n\geq 6$. In this paper we refine the result by describing all $n$vertex $k$critical graphs $G$ with $E(G)= \frac{(k+1)(k2)V(G)k(k3)}{2(k1)}$. In particular, this result implies exact values of $f_5(n)$ when $n\geq 7$.
 Publication:

arXiv eprints
 Pub Date:
 August 2014
 DOI:
 10.48550/arXiv.1408.0846
 arXiv:
 arXiv:1408.0846
 Bibcode:
 2014arXiv1408.0846K
 Keywords:

 Mathematics  Combinatorics;
 05C15;
 05C35
 EPrint:
 arXiv admin note: text overlap with arXiv:1209.1050