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Saturated graphs with minimal number of edges
TLDR
The minimal number of edges in F-saturated graphs is examined and general estimations are given and the structure of minimal graphs is described for some special forbidden graphs (stars, paths, m pairwise disjoint edges).
Complexity of Coloring Graphs without Forbidden Induced Subgraphs
We give a complete characterization of parameter graphs H for which the problem of coloring H-free graphs is polynomial and for which it is NP-complete. We further initiate a study of this problem
One More Occurrence of Variables Makes Satisfiability Jump From Trivial to NP-Complete
TLDR
It is proved that for every $k \geqslant 3$ there is an integer $f(k)$ such that $(k,s)$–${\text{SAT}}$ is trivial for $s \leqSlant f( k)$ and is NP-complete for £s \geQslant f (k) + 1$.
Dominating cliques in P5-free graphs
All induced connected subgraphs of a graphG contain a dominating set of pair-wise adjacent vertices if and only if there is no induced subgraph isomorphic to a path or a cycle of five vertices inG.
Radius, diameter, and minimum degree
On Rainbow Connection
TLDR
This paper proves several non-trivial upper bounds for $rc(G)$, as well as determine sufficient conditions that guarantee that if $G$ is a connected graph with $n$ vertices and with minimum degree $3$ then $rc (G)=2$.
Improved lower bounds on k-independence
TLDR
A lower bound for the maximum cardinality of a k-independent set—in terms of degree sequences—is proved which strengthens and generalizes several previously known results, including Turan's theorem.
Brooks-type theorems for choosability with separation
TLDR
Every graph of maximum degree admits a list coloring for every such list assignment, provided p p 5:437c, and the smallest value of p is shown to have asymptotics (1+o(1))) p n.
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