Adding Edges to Increase the Chromatic Number of a Graph. (31st March 2016)

• Record Type: Journal Article
• Authors: KOSTOCHKA, ALEXANDR; NEŠETŘIL, JAROSLAV
• Is Part Of: Combinatorics, probability and computing. Volume 25:Number 4(2016:Jul.)
• Page Start: 592
• Page End: 594
• Publication Date: 2016-03-31
• Subjects: Primary 05C15, -- Secondary 05C35; Combinatorial analysis -- Periodicals; Probabilities -- Periodicals; Computer science -- Mathematics -- Periodicals; 511.6
• DOI: 10.1017/S0963548316000146
• Languages: English
• ISSNs: 0963-5483

Abstract

Abstract : If n ⩾ k + 1 and G is a connected n -vertex graph, then one can add $\binom{k}{2}$ edges to G so that the resulting graph contains the complete graph K k +1 . This yields that for any connected graph G with at least k + 1 vertices, one can add $\binom{k}{2}$ edges to G so that the resulting graph has chromatic number > k . A long time ago, Bollobás suggested that for every k ⩾ 3 there exists a k -chromatic graph Gk such that after adding to it any $\binom{k}{2}$ − 1 edges, the chromatic number of the resulting graph is still k . In this note we prove this conjecture.