Rainbow domination and related problems on some classes of perfect graphs

Let $k \in \mathbb{N}$ and let $G$ be a graph. A function $f: V(G) \rightarrow 2^{[k]}$ is a rainbow function if, for every vertex $x$ with $f(x)=\emptyset$, $f(N(x)) =[k]$. The rainbow domination number $\gamma_{kr}(G)$ is the minimum of $\sum_{x \in V(G)} |f(x)|$ over all rainbow functions. We investigate the rainbow domination problem for some classes of perfect graphs

Remarks on the existence of uniquely partitionable planar graphs

We consider the problem of the existence of uniquely partitionable planar graphs. We survey some recent results and we prove the nonexistence of uniquely (D1,D1)-partitionable planar graphs with respect to the property D1 "to be a forest"

A Survey on Alliances and Related Parameters in Graphs

In this paper, we show that several graph parameters are known in different areas under completely different names.More specifically, our observations connect signed domination, monopolies, $\alpha$-domination, $\alpha$-independence,positive influence domination,and a parameter associated to fast information propagationin networks to parameters related to various notions of global $r$-alliances in graphs.We also propose a new framework, called (global) $(D,O)$-alliances, not only in order to characterizevarious known variants of alliance and domination parameters, but also to suggest a unifying framework for the study of alliances and domination.Finally, we also give a survey on the mentioned graph parameters, indicating how results transfer due to our observations

Burling graphs, chromatic number, and orthogonal tree-decompositions

A classic result of Asplund and Gr\"unbaum states that intersection graphs of axis-aligned rectangles in the plane are $\chi$-bounded. This theorem can be equivalently stated in terms of path-decompositions as follows: There exists a function $f:\mathbb{N}\to\mathbb{N}$ such that every graph that has two path-decompositions such that each bag of the first decomposition intersects each bag of the second in at most $k$ vertices has chromatic number at most $f(k)$. Recently, Dujmovi\'c, Joret, Morin, Norin, and Wood asked whether this remains true more generally for two tree-decompositions. In this note we provide a negative answer: There are graphs with arbitrarily large chromatic number for which one can find two tree-decompositions such that each bag of the first decomposition intersects each bag of the second in at most two vertices. Furthermore, this remains true even if one of the two decompositions is restricted to be a path-decomposition. This is shown using a construction of triangle-free graphs with unbounded chromatic number due to Burling, which we believe should be more widely known.Comment: v3: minor changes made following comments by the referees, v2: minor edit

THE ELECTRONIC JOURNAL OF COMBINATORICS (2014), DS1.14 References

and Computing 11. The results of 143 references depend on computer algorithms. The references are ordered alphabetically by the last name of the first author, and where multiple papers have the same first author they are ordered by the last name of the second author, etc. We preferred that all work by the same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations are not in alphabetical order. For example, [BaRT] is earlier on the list than [BaLS]. We also wish to explain a possible confusion with respect to the order of parts and spelling of Chinese names. We put them without any abbreviations, often with the last name written first as is customary in original. Sometimes this is different from the citations in other sources. One can obtain all variations of writing any specific name by consulting the authors database of Mathematical Reviews a

Bilangan Keterhubungan Pelangi Sejati Dari Graf

Pewarnaan-sisi pada graf G adalah suatu fungsi W∶E(G)→{1,2,…,k}=[k] di mana [k] adalah himpunan warna. Pewarnaan-sisi-sejati pada graf G merupakan pewarnaan-sisi G di mana setiap dua sisi yang terkait pada titik yang sama berwarna berbeda (Budayasa, 2007). Subgraf H dari G dengan pewarnaan W dikatakan pelangi apabila seluruh sisi H mendapat warna yang berbeda-beda. Graf G dikatakan terhubung pelangi apabila untuk setiap dua titik G, ada lintasan pelangi yang menghubungkan kedua titik tersebut. Bilangan keterhubungan pelangi graf G adalah minimum banyaknya warna yang diperlukan agar G terhubung pelangi, disimbolkan dengan rc(G). Graf nontrivial G dengan pewarnaan-sisi-sejati dikatakan terhubung pelangi sejati apabila untuk setiap dua titik yang berbeda di graf G ada lintasan pelangi yang mengaitkan dua titik tersebut. Bilangan keterhubungan pelangi sejati graf G disimbolkan dengan prc(G). Dalam pembahasan artikel ini, akan ditunjukkan bilangan keterhubungan pelangi sejati pada Graf Pohon (Tn), Graf Sikel (Cn), dan Graf Komplet (Kn). Selain itu, akan ditunjukkan juga batas atas dan batas bawah bilangan keterhubungan pelangi sejati pada graf, besarnya selisih prc(G)-rc(G), dan kelas graf dengan prc(G)=χ'(G).Kata Kunci: Graf, Pewarnaan-Sisi-Sejati Graf, Bilangan Keterhubungan Pelangi Sejati