3 research outputs found
Relations between global forcing number and maximum anti-forcing number of a graph
The global forcing number of a graph G is the minimal cardinality of an edge
subset discriminating all perfect matchings of G, denoted by gf(G). For any
perfect matching M of G, the minimal cardinality of an edge subset S in E(G)-M
such that G-S has a unique perfect matching is called the anti-forcing number
of M,denoted by af(G, M). The maximum anti-forcing number of G among all
perfect matchings is denoted by Af(G). It is known that the maximum
anti-forcing number of a hexagonal system equals the famous Fries number.
We are interested in some comparisons between the global forcing number and
the maximum anti-forcing number of a graph. For a bipartite graph G, we show
that gf(G)is larger than or equal to Af(G). Next we mainly extend such result
to non-bipartite graphs, which is the set of all graphs with a perfect matching
which contain no two disjoint odd cycles such that their deletion results in a
subgraph with a perfect matching. For any such graph G, we also have gf(G) is
larger than or equal to Af(G) by revealing further property of non-bipartite
graphs with a unique perfect matching. As a consequence, this relation also
holds for the graphs whose perfect matching polytopes consist of non-negative
1-regular vectors. In particular, for a brick G, de Carvalho, Lucchesi and
Murty [4] showed that G satisfying the above condition if and only if G is
solid, and if and only if its perfect matching polytope consists of
non-negative 1-regular vectors.
Finally, we obtain tight upper and lower bounds on gf(G)-Af(G). For a
connected bipartite graph G with 2n vertices, we have that 0 \leq gf(G)-Af(G)
\leq 1/2 (n-1)(n-2); For non-bipartite case, -1/2 (n^2-n-2) \leq gf(G)-Af(G)
\leq (n-1)(n-2).Comment: 19 pages, 11 figure
An aperiodic monotile
A longstanding open problem asks for an aperiodic monotile, also known as an
"einstein": a shape that admits tilings of the plane, but never periodic
tilings. We answer this problem for topological disk tiles by exhibiting a
continuum of combinatorially equivalent aperiodic polygons. We first show that
a representative example, the "hat" polykite, can form clusters called
"metatiles", for which substitution rules can be defined. Because the metatiles
admit tilings of the plane, so too does the hat. We then prove that generic
members of our continuum of polygons are aperiodic, through a new kind of
geometric incommensurability argument. Separately, we give a combinatorial,
computer-assisted proof that the hat must form hierarchical -- and hence
aperiodic -- tilings.Comment: 89 pages, 57 figures; Minor corrections, renamed "fylfot" to
"triskelion", added the name "turtle", added references, new H7/H8 rules (Fig
2.11), talk about reflection