685 research outputs found
Triangle-free subgraphs of random graphs
Recently there has been much interest in studying random graph analogues of
well known classical results in extremal graph theory. Here we follow this
trend and investigate the structure of triangle-free subgraphs of with
high minimum degree. We prove that asymptotically almost surely each
triangle-free spanning subgraph of with minimum degree at least
is -close to bipartite,
and each spanning triangle-free subgraph of with minimum degree at
least is -close to
-partite for some . These are random graph analogues of a
result by Andr\'asfai, Erd\H{o}s, and S\'os [Discrete Math. 8 (1974), 205-218],
and a result by Thomassen [Combinatorica 22 (2002), 591--596]. We also show
that our results are best possible up to a constant factor.Comment: 18 page
Bounds for the smallest k-chromatic graphs of given girth
Let n(g)(k) denote the smallest order of a k-chromatic graph of girth at least g. We consider the problem of determining n(g)(k) for small values of k and g. After giving an overview of what is known about n(g)(k), we provide some new lower bounds based on exhaustive searches, and then obtain several new upper bounds using computer algorithms for the construction of witnesses, and for the verification of their correctness. We also present the first examples of reasonably small order for k = 4 and g > 5. In particular, the new bounds include: n(4)(7) <= 77, 26 <= n(6)(4) <= 66 and 30 <= n(7)(4) <= 171
Uniquely D-colourable digraphs with large girth
Let C and D be digraphs. A mapping is a C-colouring if for
every arc of D, either is an arc of C or , and the
preimage of every vertex of C induces an acyclic subdigraph in D. We say that D
is C-colourable if it admits a C-colouring and that D is uniquely C-colourable
if it is surjectively C-colourable and any two C-colourings of D differ by an
automorphism of C. We prove that if a digraph D is not C-colourable, then there
exist digraphs of arbitrarily large girth that are D-colourable but not
C-colourable. Moreover, for every digraph D that is uniquely D-colourable,
there exists a uniquely D-colourable digraph of arbitrarily large girth. In
particular, this implies that for every rational number , there are
uniquely circularly r-colourable digraphs with arbitrarily large girth.Comment: 21 pages, 0 figures To be published in Canadian Journal of
Mathematic
Triangle-free subgraphs of random graphs
Recently there has been much interest in studying random graph analogues of well known classical results in extremal graph theory. Here we follow this trend and investigate the structure of triangle-free subgraphs of G(n, p) with high minimum degree. We prove that asymptotically almost surely each triangle-free spanning subgraph of G(n, p) with minimum degree at least ( 2 + o(1)lpn is O(p−1 n)-close to bipartite, and each spanning triangle-free subgraph of G(n, p) with minimum degree at least ( 1 + ε)pn is O(p−1 n)-close to r-partite for some r = r(ε). These are random graph analogues of a result by Andrásfai, Erdős and Sós [Discrete Math. 8 (1974), 205–218], and a result by Thomassen [Combinatorica 22 (2002), 591–596]. We also show that our results are best possible up to a constant factor
A tamed family of triangle-free graphs with unbounded chromatic number
We construct a hereditary class of triangle-free graphs with unbounded
chromatic number, in which every non-trivial graph either contains a pair of
non-adjacent twins or has an edgeless vertex cutset of size at most two. This
answers in the negative a question of Chudnovsky, Penev, Scott, and Trotignon.
The class is the hereditary closure of a family of (triangle-free) twincut
graphs such that has chromatic number . We also
show that every twincut graph is edge-critical
A survey of -boundedness
If a graph has bounded clique number, and sufficiently large chromatic
number, what can we say about its induced subgraphs? Andr\'as Gy\'arf\'as made
a number of challenging conjectures about this in the early 1980's, which have
remained open until recently; but in the last few years there has been
substantial progress. This is a survey of where we are now
- …