Bounding Tree-Width via Contraction on the Projective Plane and Torus.

If $X$ is a collection of edges in a graph $G$, let $G/X$ denote the contraction of $X$. Following a question of Oxley and a conjecture of Oporowski, we prove that every projective planar graph $G$ admits an edge partition $\{X,Y\}$ such that $G/X$ and $G/Y$ have tree-width at most three. We prove that every toroidal graph $G$ admits an edge partition $\{X,Y\}$ such that $G/X$ and $G/Y$ have tree-width at most three and four, respectively.

A BOUND ON THE BONDAGE NUMBER OF TOROIDAL GRAPHS

The bondage number b(G) of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph with domination number greater than γ(G), where γ(G) denote the domination number of G. Let G be a toroidal graph with maximum degree Δ(G). In this paper, we show that b(G) ≤ 9. Moveover, if Δ(G) ≠ 6, then b(G) ≤ 8.

ON (3, 1)*-CHOOSABILITY OF TOROIDAL GRAPHS

A list-assignment L to the vertices of G is an assignment of a set L(v) of colors to vertex v for every v ∈ V(G). Given a list-assignment L and integer d ≥ 0, an (L, d)*-coloring of a graph G is a mapping ϕ that assigns a color ϕ(v) ∈ L(v) to each vertex v ∈ V(G) such that at most d neighbors of v receive color ϕ(v). A graph is called (k,d)*-choosable, if G admits an (L, d)*-coloring for every list assignment L with |L(v)| ≥ k for all v ∈ V(G). In this note, it is proved that every toroidal graph, which contains neither i-cycles nor j-cycles for any subset {i, j} ⊆ {3, 4, 6}, is (3, 1)*-choosable.

INCOMPRESSIBLE SURFACES AND (1, 1)-KNOTS

Let M be S3, S1 × S2, or a lens space L(p, q), and let k be a (1, 1)-knot in M. We show that if there is a closed meridionally incompressible surface in the complement of k, then the surface and the knot can be put in a special position, namely, the surface is the boundary of a regular neighborhood of a toroidal graph, and the knot is level with respect to that graph. As an application we show that for any such M there exist tunnel number one knots which are not (1, 1)-knots.