5,821 research outputs found
Spanning Eulerian subgraphs and Catlin’s reduced graphs
A graph G is collapsible if for every even subset R ⊆ V (G), there is a spanning connected subgraph HR of G whose set of odd degree vertices is R. A graph is reduced if it has no nontrivial collapsible subgraphs. Catlin [4] showed that the existence of spanning Eulerian subgraphs in a graph G can be determined by the reduced graph obtained from G by contracting all the collapsible subgraphs of G. In this paper, we present a result on 3-edge-connected reduced graphs of small orders. Then, we prove that a 3-edge-connected graph G of order n either has a spanning Eulerian subgraph or can be contracted to the Petersen graph if G satisfies one of the following:
(i) d(u) + d(v) \u3e 2(n/15 − 1) for any uv 6∈ E(G) and n is large;
(ii) the size of a maximum matching in G is at most 6;
(iii) the independence number of G is at most 5.
These are improvements of prior results in [16], [18], [24] and [25]
Properties of Catlin's reduced graphs and supereulerian graphs
A graph is called collapsible if for every even subset ,
there is a spanning connected subgraph of such that is the set of
vertices of odd degree in . A graph is the reduction of if it is
obtained from by contracting all the nontrivial collapsible subgraphs. A
graph is reduced if it has no nontrivial collapsible subgraphs. In this paper,
we first prove a few results on the properties of reduced graphs. As an
application, for 3-edge-connected graphs of order with for any where are given, we show how such graphs
change if they have no spanning Eulerian subgraphs when is increased from
to 10 then to
High Chern number quantum anomalous Hall phases in graphene ribbons with Haldane orbital coupling
We investigate possible phase transitions among the different quantum
anomalous Hall (QAH) phases in a zigzag graphene ribbon under the influence of
the exchange field. The effective tight-binding Hamiltonian for graphene is
made up of the hopping term, the Kane-Mele and Rashba spin-orbit couplings as
well as the Haldane orbital term. We find that the variation of the exchange
field results in bulk gap-closing phenomena and phase transitions occur in the
graphene system. If the Haldane orbital coupling is absent, the phase
transition between the chiral (anti-chiral) edge state () and
the pseudo-quantum spin Hall state () takes place. Surprisingly, when
the Haldane orbital coupling is taken into account, an intermediate QSH phase
with two additional edge modes appears in between phases and .
This intermediate phase is therefore either the hyper-chiral edge state of high
Chern number or anti-hyper-chiral edge state of when the
direction of exchange field is reversed. We present the band structures, edge
state wave functions and current distributions of the different QAH phases in
the system. We also report the critical exchange field values for the QAH phase
transitions.Comment: 4 figure
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