http://users.uoa.gr/~sedthilk/papers/abgs.pdf

An annotated bibliography on guaranteed graph searching

We prove a conjecture of Favaron et al. that every graph of order n and minimum degree at least three has a total dominating set of size at least n-2. We also present several related results about: (1) extentions to graphs of minimum degree two, (2) examining graphs where the bound is tight, and (3) a type of bipartite domination and its relation to transversals in hypergraphs. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 207–210, 2004

/pdf/some-remarks-on-domination-30clcms7q1.pdf

Some remarks on domination

https://people.cs.clemson.edu/~goddard/papers/idomSurvey.pdf

Independent domination in graphs: A survey and recent results

What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F-packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect F-packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect F-packing. 
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy and Szemeredi have proved to be powerful tools in attacking such problems and quite recently, several long-standing problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on F-packings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved.

Embedding large subgraphs into dense graphs

For a graph $H$ and an integer $n$, the Turan number $\ex(n,H)$ is the maximum possible number of edges in a simple graph on $n$ vertices that contains no copy of $H$. $H$ is $r$-degenerate if every one of its subgraphs contains a vertex of degree at most $r$. We prove that, for any fixed bipartite graph $H$ in which all degrees in one colour class are at most $r$, $\ex(n,H)\,{\leq}\,O(n^{2-1/r})$. This is tight for all values of $r$ and can also be derived from an earlier result of Furedi. We also show that there is an absolute positive constant $c$ such that, for every fixed bipartite $r$-degenerate graph $H$, $\ex(n,H)\,{\leq}\,O(n^{1-c/r}).$ This is motivated by a conjecture of Erdos that asserts that, for every such $H$, $\ex(n,H)\,{\leq}\,O(n^{1-1/r}).$For two graphs $G$ and $H$, the Ramsey number $r(G,H)$ is the minimum number $n$ such that, in any colouring of the edges of the complete graph on $n$ vertices by red and blue, there is either a red copy of $G$ or a blue copy of $H$. Erdos conjectured that there is an absolute constant $c$ such that, for any graph $G$ with $m$ edges, $r(G,G)\,{\leq}\,2^{c \sqrt m}$. Here we prove this conjecture for bipartite graphs $G$, and prove that for general graphs $G$ with $m$ edges, $r(G,G)\,{\leq}\,2^{c \sqrt m \log m}$ for some absolute positive constant $c$.These results and some related ones are derived from a simple and yet surprisingly powerful lemma, proved, using probabilistic techniques, at the beginning of the paper. This lemma is a refined version of earlier results proved and applied by various researchers including Rodl, Kostochka, Gowers and Sudakov.

/pdf/turan-numbers-of-bipartite-graphs-and-related-ramsey-type-3ttdd47k8a.pdf

Turán Numbers of Bipartite Graphs and Related Ramsey-Type Questions

Directed Tree-Width

We show that if G is a simple connected graph with $$|E\;({\bf G})|\geq |V\;(\,G)|+{1 \over 2}t\,\;(t-1)$$ and $|V(G)| \,\neq\,t+2$, then G has a spanning tree with > t leaves, and this is best possible. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 189–197, 2001

Spanning trees with many leaves

Generating Internally Four-Connected Graphs

In [Directed tree-width, J. Combin. Theory Ser. B 82 (2001), 138-154] we introduced the notion of tree-width of directed graphs and presented a conjecture, formulated during discussions with Noga Alon and Bruce Reed, stating that a digraph of huge tree-width has a large "cylindrical grid" minor. Here we prove the conjecture for planar digraphs, but many steps of the proof work in general. 
This is an unedited and unpolished manuscript from October 2001. Since many people asked for copies we are making it available in the hope that it may be useful. The conjecture was proved by Kawarabayashi and Kreutzer in arXiv:1411.5681.

Excluding A Grid Minor In Planar Digraphs.

We prove that for every 3-edge-connected graph G there exists a partition of E(G) into at most nine sets {X1, X2, . . . , Xm} so that G \Xi is 2-edge-connected for every 1 ≤ i ≤ m. We then generalize this result, proving that for every (2k+1)-edgeconnected graph G there is a partition of E(G) into a bounded number (depending only on k) of sets {X1, X2, . . . , Xm} so that G \ Xi is 2k-edge-connected for every 1 ≤ i ≤ m. In the third section of the paper, we apply our theorem to prove that for every 3edge-connected graph G and every map p : E(G) → {84, 86}, there exists a multiset of circuits of G so that every edge e ∈ E(G) is contained in exactly p(e) of these circuits. This resolves a question of Goddyn. This result is then bootstrapped to a more general theorem on circuit covers. If G = (V,E) is a directed graph, a flow φ : E → Z of G is called an antisymmetric flow if φ(E) ∩ −φ(E) = ∅. In the fourth section of the paper, we apply our theorem to show that every directed graph with no directed edge-cuts of size ≤ 2 has an antisymmetric flow of bounded size. This resolves a question of Nesetřil and Raspaud. ∗This research was supported by Navy Aasert grant N00014-98-1-0457 †This research was supported by ‡This research was supported by the ONR under grant N0014-97-1-0512 and the NSF under grant DMS 9701598

Cut Coloring and Circuit Covering

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