2,107 research outputs found
Embedding graphs having Ore-degree at most five
Let and be graphs on vertices, where is sufficiently large.
We prove that if has Ore-degree at most 5 and has minimum degree at
least then Comment: accepted for publication at SIAM J. Disc. Mat
Cohen-Macaulay graphs and face vectors of flag complexes
We introduce a construction on a flag complex that, by means of modifying the
associated graph, generates a new flag complex whose -factor is the face
vector of the original complex. This construction yields a vertex-decomposable,
hence Cohen-Macaulay, complex. From this we get a (non-numerical)
characterisation of the face vectors of flag complexes and deduce also that the
face vector of a flag complex is the -vector of some vertex-decomposable
flag complex. We conjecture that the converse of the latter is true and prove
this, by means of an explicit construction, for -vectors of Cohen-Macaulay
flag complexes arising from bipartite graphs. We also give several new
characterisations of bipartite graphs with Cohen-Macaulay or Buchsbaum
independence complexes.Comment: 14 pages, 3 figures; major updat
Realizations of self branched coverings of the 2-sphere
For a degree d self branched covering of the 2-sphere, a notable
combinatorial invariant is an integer partition of 2d -- 2, consisting of the
multiplicities of the critical points. A finer invariant is the so called
Hurwitz passport. The realization problem of Hurwitz passports remain largely
open till today. In this article, we introduce two different types of finer
invariants: a bipartite map and an incident matrix. We then settle completely
their realization problem by showing that a map, or a matrix, is realized by a
branched covering if and only if it satisfies a certain balanced condition. A
variant of the bipartite map approach was initiated by W. Thurston. Our results
shed some new lights to the Hurwitz passport problem
Minimum Degrees of Minimal Ramsey Graphs for Almost-Cliques
For graphs and , we say is Ramsey for if every -coloring of
the edges of contains a monochromatic copy of . The graph is Ramsey
-minimal if is Ramsey for and there is no proper subgraph of
so that is Ramsey for . Burr, Erdos, and Lovasz defined to
be the minimum degree of over all Ramsey -minimal graphs . Define
to be a graph on vertices consisting of a complete graph on
vertices and one additional vertex of degree . We show that
for all values ; it was previously known that , so it
is surprising that is much smaller.
We also make some further progress on some sparser graphs. Fox and Lin
observed that for all graphs , where is
the minimum degree of ; Szabo, Zumstein, and Zurcher investigated which
graphs have this property and conjectured that all bipartite graphs without
isolated vertices satisfy . Fox, Grinshpun, Liebenau,
Person, and Szabo further conjectured that all triangle-free graphs without
isolated vertices satisfy this property. We show that -regular -connected
triangle-free graphs , with one extra technical constraint, satisfy ; the extra constraint is that has a vertex so that if one
removes and its neighborhood from , the remainder is connected.Comment: 10 pages; 3 figure
Ore-degree threshold for the square of a Hamiltonian cycle
A classic theorem of Dirac from 1952 states that every graph with minimum
degree at least n/2 contains a Hamiltonian cycle. In 1963, P\'osa conjectured
that every graph with minimum degree at least 2n/3 contains the square of a
Hamiltonian cycle. In 1960, Ore relaxed the degree condition in the Dirac's
theorem by proving that every graph with for every contains a Hamiltonian cycle. Recently, Ch\^au proved an Ore-type
version of P\'osa's conjecture for graphs on vertices using the
regularity--blow-up method; consequently the is very large (involving a
tower function). Here we present another proof that avoids the use of the
regularity lemma. Aside from the fact that our proof holds for much smaller
, we believe that our method of proof will be of independent interest.Comment: 24 pages, 1 figure. In addition to some fixed typos, this updated
version contains a simplified "connecting lemma" in Section 3.
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