293 research outputs found
Degree Sequences and the Existence of -Factors
We consider sufficient conditions for a degree sequence to be forcibly
-factor graphical. We note that previous work on degrees and factors has
focused primarily on finding conditions for a degree sequence to be potentially
-factor graphical.
We first give a theorem for to be forcibly 1-factor graphical and, more
generally, forcibly graphical with deficiency at most . These
theorems are equal in strength to Chv\'atal's well-known hamiltonian theorem,
i.e., the best monotone degree condition for hamiltonicity. We then give an
equally strong theorem for to be forcibly 2-factor graphical.
Unfortunately, the number of nonredundant conditions that must be checked
increases significantly in moving from to , and we conjecture that
the number of nonredundant conditions in a best monotone theorem for a
-factor will increase superpolynomially in .
This suggests the desirability of finding a theorem for to be forcibly
-factor graphical whose algorithmic complexity grows more slowly. In the
final section, we present such a theorem for any , based on Tutte's
well-known factor theorem. While this theorem is not best monotone, we show
that it is nevertheless tight in a precise way, and give examples illustrating
this tightness.Comment: 19 page
On the stable degree of graphs
We define the stable degree s(G) of a graph G by s(G)â=â min max d (v), where the minimum is taken over all maximal independent sets U of G. For this new parameter we prove the following. Deciding whether a graph has stable degree at most k is NP-complete for every fixed kââ„â3; and the stable degree is hard to approximate. For asteroidal triple-free graphs and graphs of bounded asteroidal number the stable degree can be computed in polynomial time. For graphs in these classes the treewidth is bounded from below and above in terms of the stable degree
A Survey of Best Monotone Degree Conditions for Graph Properties
We survey sufficient degree conditions, for a variety of graph properties,
that are best possible in the same sense that Chvatal's well-known degree
condition for hamiltonicity is best possible.Comment: 25 page
On a Directed Tree Problem Motivated by a Newly Introduced Graph Product
In this paper we introduce and study a directed tree problem motivated by a new graph product that we have recently introduced and analysed in two conference contributions in the context of periodic real-time processes. While the two conference papers were focussing more on the applications, here we mainly deal with the graph theoretical and computational complexity issues. We show that the directed tree problem is NP-complete and present and compare several heuristics for this problem
Forbidden subgraphs that imply hamiltonian-connectedness
It is proven that if G is a 3âconnected clawâfree graph which is also H1âfree (where H1 consists of two disjoint triangles connected by an edge), then G is hamiltonianâconnected. Also, examples will be described that determine a finite family of graphs equation image such that if a 3âconnected graph being clawâfree and Lâfree implies G is hamiltonianâconnected, then L equation imag
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