Decomposition theorem on matchable distributive lattices

A distributive lattice structure ${\mathbf M}(G)$ has been established on the set of perfect matchings of a plane bipartite graph $G$. We call a lattice {\em matchable distributive lattice} (simply MDL) if it is isomorphic to such a distributive lattice. It is natural to ask which lattices are MDLs. We show that if a plane bipartite graph $G$ is elementary, then ${\mathbf M}(G)$ is irreducible. Based on this result, a decomposition theorem on MDLs is obtained: a finite distributive lattice $\mathbf{L}$ is an MDL if and only if each factor in any cartesian product decomposition of $\mathbf{L}$ is an MDL. Two types of MDLs are presented: $J(\mathbf{m}\times \mathbf{n})$ and $J(\mathbf{T})$, where $\mathbf{m}\times \mathbf{n}$ denotes the cartesian product between $m$-element chain and $n$-element chain, and $\mathbf{T}$ is a poset implied by any orientation of a tree.Comment: 19 pages, 7 figure

On the semi-proper orientations of graphs

International audienceA weighted orientation of a graph G is a pair (D, w) where D is an orientation of G and w is an arc-weighting of D, that is an application A(D) → N \ {0}. The in-weight of a vertex v in a weighted orientation (D, w), denoted by S (D,w) (v), is the sum of the weights of arcs with head v in D. A semi-proper orientation is a weighted orientation such that two adjacent vertices have different in-weights. The semiproper orientation number of a graph G, denoted by − → χs(G), is min (D,w)∈Γ max v∈V (G) S (D,w) (v), where Γ is the set of all semi-proper orientations of G. A semi-proper orientation (D, w) of a graph G is optimal if max v∈V (G) S (D,w) (v) = − → χs(G). In this work, we show that every graph G has an optimal semi-proper orientation (D, w) such that the weight of each arc is 1 or 2. We then give some bounds on the semi-proper orientation number: we show Mad(G) 2 ≤ − → χs(G) ≤ Mad(G) 2 +χ(G)−1 and δ * (G)+1 2 ≤ − → χs(G) ≤ 2δ * (G) for all graph G, where Mad(G) and δ * (G) are the maximum average degree and the degeneracy of G, respectively. We then deduce that the maximum semi-proper orientation number of a tree is 2, of a cactus is 3, of an outerplanar graph is 4, and of a planar graph is 6. Finally, we consider the computational complexity of associated problems: we show that determining whether − → χs(G) = χ(G) is NP-complete for planar graphs G with − → χs(G) = 2; we also show that deciding whether − → χs(G) ≤ 2 is NP-complete for planar bipartite graphs G

4-colored graphs and knot/link complements

A representation for compact 3-manifolds with non-empty non-spherical boundary via 4-colored graphs (i.e., 4-regular graphs endowed with a proper edge-coloration with four colors) has been recently introduced by two of the authors, and an initial classification of such manifolds has been obtained up to 8 vertices of the representing graphs. Computer experiments show that the number of graphs/manifolds grows very quickly as the number of vertices increases. As a consequence, we have focused on the case of orientable 3-manifolds with toric boundary, which contains the important case of complements of knots and links in the 3-sphere. In this paper we obtain the complete catalogation/classification of these 3-manifolds up to 12 vertices of the associated graphs, showing the diagrams of the involved knots and links. For the particular case of complements of knots, the research has been extended up to 16 vertices.Comment: 19 pages, 6 figures, 3 tables; changes in Lemma 6, Corollaries 7 and