On small uniquely vertex-colourable graphs and Xu's conjecture

• Published in: Discrete mathematics Vol. 223; no. 1; pp. 93 - 108
• Main Authors: Daneshgar, Amir, Naserasr, Reza
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 28-08-2000
• Subjects: Graph colouring; Uniquely vertex-colourable graphs; Xu's conjecture; Graph colouring; Xu's conjecture; Uniquely vertex-colourable graphs
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/S0012-365X(00)00042-X

Consider the parameter
Λ(G)=|E(G)|−|V(G)|(k−1)+(
k
2
)
for a k-chromatic graph G, on the set of vertices
V(
G) and with the set of edges
E(
G). It is known that
Λ(
G)⩾0 for any k-chromatic uniquely vertex-colourable graph G (k-UCG), and, S.J. Xu has conjectured that for any k-UCG,
G,
Λ(G)=0
implies that cl(
G)=
k; in which cl(
G) is the clique number of G. In this paper, first, we introduce the concept of the
core of a k-UCG as an induced subgraph without any colour-class of size one, and without any vertex of degree
k−1. Considering (
k,
n)-cores as k-UCGs on n vertices, we show that edge-minimal (
k,2
k)-cores do not exist when
k⩾3, which shows that for any edge-minimal k-UCG on 2
k vertices either the conjecture is true or there exists a colour-class of size one. Also, we consider the structure of edge-minimal (
k,2
k+1)-cores and we show that such cores exist for all
k⩾4. Moreover, we characterize all edge-minimal (4,9)-cores and we show that there are only seven such cores (up to isomorphism). Our proof shows that Xu's conjecture is true in the case of edge-minimal (4,9)-cores.