Certain subgraphs of a given graph G restrict the minimum number χ(G) of colors that can be assigned to the vertices of G such that the endpoints of all edges receive distinct colors. Some of such subgraphs are related to the celebrated Strong Perfect Graph Theorem, as it implies that every graph G contains a clique of size χ(G), or an odd hole or an odd anti-hole as an induced subgraph. In thi...

We present a simple and e cient dictionary with worst case constant lookup time, equaling the theoretical performance of the classic dynamic perfect hashing scheme of Dietzfelbinger et al. (Dynamic perfect hashing: Upper and lower bounds. SIAM J. Comput., 23(4):738 761, 1994). The space usage is similar to that of binary search trees, i.e., three words per key on average. The practicality of th...

A graph is Berge if no induced subgraph of it is an odd cycle of length at least five or the complement of one. In joint work with Robertson, Seymour, and Thomas we recently proved the Strong Perfect Graph Theorem, which was a conjecture about the chromatic number of Berge graphs. The proof consisted of showing that every Berge graph either belongs to one of a few basic classes, or admits one o...

Normal graphs are defined in terms of cross-intersecting set families: a graph is normal if it admits a clique cover Q and a stable set cover S s.t. every clique in Q intersects every stable set in S. Normal graphs can be considered as closure of perfect graphs by means of co-normal products (K ̈orner [6]) and graph entropy (Czisz ́ar et al. [5]). Perfect graphs have been recently characterized a...

The ability to write a research article is one of students’ obligation that must be mastered because final year university students are always faced the project especially writing an published in national journal as requirement pass or graduate from campus where she/he lectures. This aims identify errors using tenses articles. used method qualitative descriptive with test – pre-test and post-te...

This study investigated common English writing errors made by undergraduate ESL learners at Aligarh Muslim University whose first language is Urdu-Hindi. The carried out with the help of written compositions from 20 participants. data were analyzed within a broader framework propounded S. Pit Corder. researcher concluded that study's in form spelling, copula, word choice, prepositions, verb ten...