the tenacity of a graph g, t(g), is de ned by t(g) = min{[|s|+τ(g-s)]/[ω(g-s)]}, where the minimum is taken over all vertex cutsets s of g. we de ne τ(g - s) to be the number of the vertices in the largest component of the graph g - s, and ω(g - s) be the number of components of g - s.in this paper a lower bound for the tenacity t(g) of a graph with genus γ(g) is obtained using the graph's...

The tenacity of a graph G, T(G), is dened by T(G) = min{[|S|+τ(G-S)]/[ω(G-S)]}, where the minimum is taken over all vertex cutsets S of G. We dene τ(G - S) to be the number of the vertices in the largest component of the graph G - S, and ω(G - S) be the number of components of G - S.In this paper a lower bound for the tenacity T(G) of a graph with genus γ(G) is obtained using the graph's connec...

the vertex arboricity $rho(g)$ of a graph $g$ is the minimum number of subsets into which the vertex set $v(g)$ can be partitioned so that each subset induces an acyclic graph‎. ‎a graph $g$ is called list vertex $k$-arborable if for any set $l(v)$ of cardinality at least $k$ at each vertex $v$ of $g$‎, ‎one can choose a color for each $v$ from its list $l(v)$ so that the subgraph induced by ev...

The vertex arboricity $rho(G)$ of a graph $G$ is the minimum number of subsets into which the vertex set $V(G)$ can be partitioned so that each subset induces an acyclic graph‎. ‎A graph $G$ is called list vertex $k$-arborable if for any set $L(v)$ of cardinality at least $k$ at each vertex $v$ of $G$‎, ‎one can choose a color for each $v$ from its list $L(v)$ so that the subgraph induced by ev...

We study the toroidality testing and toroidal embedding problems in the context of bounded space algorithms. Specifically, we show that the problem of embedding a toroidal graph on a surface of genus 1 can be solved by an algorithm that runs in deterministic logarithmic space. The algorithm rejects whenever the graph is not toroidal. This algorithm is optimal as we also show a matching hardness...

In this paper, a structural theorem about toroidal graphs is given that strengthens a result of Borodin on plane graphs. As a consequence, it is proved that every toroidal graph without adjacent triangles is (4, 1)∗-choosable. This result is best possible in the sense that K7 is a non-(3, 1)∗-choosable toroidal graph. A linear time algorithm for producing such a coloring is presented also. © 20...

Erdős et al. (Canad. J. Math. 18 (1966) 106–112) conjecture that there exists a constant dce such that every simple graph on n vertices can be decomposed into at most dcen circuits and edges. We consider toroidal graphs, where the graphs can be embedded on the torus, and give a polynomial time algorithm to decompose the edge set of an even toroidal graph on n vertices into at most (n + 3)/2 cir...

In this paper we give a linear algorithm to edge partition a toroidal graph, i.e., graph that can be embedded on the orientable surface of genus one without edge crossing, into three forests plus a set of at most three edges. For triangulated toroidal graphs, this algorithm gives a linear algorithm for finding three edge-disjoint spanning trees. This is in a certain way an extension of the well...

1 We investigate the toroidal expanse of an embedded graph G, that is, the size of the largest 2 toroidal grid contained in G as a minor. In the course of this work we introduce a new embedding 3 density parameter, the stretch of an embedded graph G, and use it to bound the toroidal 4 expanse from above and from below within a constant factor depending only on the genus and 5 the maximum degree...

A toroidal periodic graph G D is defined by an integral d = d matrix D and a directed graph G in which the edges are associated with d-dimensional integral vectors. The periodic graph has a vertex for each vertex of the static graph and for each integral position in the parallelpiped defined by the columns of D. There is an edge from vertex u at position y to vertex ̈ at position z in the period...