Discrete Mathematics 307 (2007) 108 – 114
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A characterization of maximal non-k-factor-critical graphs
N. Ananchuen
a,1
, L. Caccetta
b,2
, W. Ananchuen
c,3
a
Department of Mathematics, Silpakorn University, Nakorn Pathom 73000, Thailand
b
Western Australian Centre of Excellence in Industrial Optimisation, Department of Mathematics and Statistics, Curtin University of Technology,
GPO Box U1987, Perth 6845, WA, Australia
c
School of Liberal Arts, Sukhothai Thammathirat Open University, Pakkred, Nonthaburi 11120, Thailand
Received 2 August 2004; received in revised form 19 April 2006; accepted 19 May 2006
Available online 18 September 2006
Abstract
A graph G of order p is k-factor-critical,where p and k are positive integers with the same parity, if the deletion of any set of k
vertices results in a graph with a perfect matching. G is called maximal non-k-factor-critical if G is not k-factor-critical but G + e is
k-factor-critical for every missing edge e/∈ E(G). A connected graph G with a perfect matching on 2n vertices is k-extendable, for
1 k n − 1, if for every matching M of size k in G there is a perfect matching in G containing all edges of M. G is called maximal
non-k-extendable if G is not k-extendable but G + e is k-extendable for every missing edge e/∈ E(G) . A connected bipartite graph G
with a bipartitioning set (X, Y ) such that |X|=|Y |=n is maximal non-k-extendable bipartite if G is not k-extendable but G + xy is
k-extendable for any edge xy /∈ E(G) with
x ∈ X and y ∈ Y . A complete characterization of maximal non-k-factor-critical graphs,
maximal non-k-extendable graphs and maximal non-k-extendable bipartite graphs is given.
© 2006 Elsevier B.V. All rights reserved.
Keywords: Matching; k-factor-critical graphs; k-extendable graphs
1. Introduction
All graphs considered in this paper are finite, connected, loopless and have no multiple edges. For the most part our
notation and terminology follows that of Bondy and Murty [2]. Thus G is a graph with vertex set V (G), edge set E(G)
and minimum degree (G).ForV
⊆ V (G), G[V
] denotes the subgraph induced by V
. Similarly, G[E
] denotes the
subgraph induced by the edge set E
of G. N
G
(u) denotes the neighbour set of u in G and N
G
(u) the non-neighbours
of u. Note that
N
G
(u) = V (G)\(N
G
(u) ∪{u}). The join G ∨ H of disjoint graphs G and H is the graph obtained from
G ∪ H by joining each vertex of G to each vertex of H.
A matching M in G is a subset of E(G) in which no two edges have a vertex in common. A vertex v is saturated
by M if some edge of M is incident to v; otherwise v is said to be unsaturated. A matching G is perfect if it saturates
every vertex of G. For simplicity, we let V(M)denote the vertex set of the subgraph G[M] induced by M. A graph G
of order p is k-factor-critical, where p and k are positive integers with the same parity, if the deletion of any set of k
E-mail address: caccetta@cs.curtin.edu.au (L. Caccetta).
1
Work supported by the Thailand Research Fund Grant # BRG4680019.
2
Work supported by the Western Australian Centre of Excellence in Industrial Optimisation (WACEIO).
3
Work supported by the Thailand Research Fund Grant # BRG/15/2545.
0012-365X/$ - see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.disc.2006.05.036