The generalized list T -coloring is a common generalization of many graph coloring models, including classical coloring, L(p, q)-labeling, channel assignment and T -coloring. Every vertex from the input graph has a list of permitted labels. Moreover, every edge has a set of forbidden differences. We ask for such a labeling of vertices of the input graph with natural numbers, in which every vert...

Roussel and Rubio proved a lemma which is essential in the proof of the Strong Perfect Graph Theorem. We give a new short proof of the main case of this lemma. In this note, we also give a short proof of Hayward’s decomposition theorem for weakly chordal graphs, relying on a Roussel–Rubio-type lemma. We recall how Roussel–Rubio-type lemmas yield very short proofs of the existence of even pairs ...

Maire introduced the class of those graphs without long holes, for which every induced subgraph possesses a vertex whose neighbourhood has no P4. Those graphs are called slightly triangulated graphs and are proven to be perfect in [14]. In this paper, we introduce a wider class of graphs, obtained by relaxing the neighbourhood condition. We also provide polynomial algorithms to recognize and to...

We show that the following problem is NP complete: Let G be a cubic bipartite graph and f be a precoloring of a subset of edges of G using at most three colors. Can f be extended to a proper edge 3-coloring of the entire graph G? This result provides a natural counterpart to classical Holyer's result on edge 3-colorability of cubic graphs and a strengthening of results on precoloring extension ...

4-methylimidazole (4-mei) is a simple nitrogen-containing heterocyclic compound, and recently classified as group 2b by international agency for research on cancer (iarc). caramel coloring is widely applied food coloring in many food products including coffee, carbonated beverages, beer and wine. in present study, ɣ-irradiation was applied to caramel color iii to reduce the level of 4-mei witho...

We study the class of simple graphs G for which every pair of distinct odd cycles intersect in at most one edge. We give a structural characterization of the graphs in G and prove that every G ∈ G satisfies the list-edge-coloring conjecture. When ∆(G) ≥ 4, we in fact prove a stronger result about kernel-perfect orientations in L(G) which implies that G is (m∆(G) : m)edge-choosable and ∆(G)-edge...

We will extend Reed's Semi-Strong Perfect Graph Theorem by proving that unbreakable C 5-free graphs diierent from a C 6 and its complement have unique P 4-structure.

We conjecture that for any fixed r and sufficiently large n, there is a monochromatic Hamiltonian Bergecycle in every (r − 1)-coloring of the edges of K n , the complete r-uniform hypergraph on n vertices. We prove the conjecture for r = 3, n 5 and its asymptotic version for r = 4. For general r we prove weaker forms of the conjecture: there is a Hamiltonian Berge-cycle in (r−1)/2 -colorings of...

The Wonderful Lemma, that was first proved by Roussel and Rubio, is one of the most important tools in the proof of the Strong Perfect Graph Theorem. Here we give a short proof of this lemma.

In this paper we consider properly edge-colored complete graphs, i.e. two edges with the same color cannot share an endpoint, so each color class is a matching. A proper edge-coloring is a factorization if each color class is a perfect or near perfect matching. A subgraph is called rainbow if its edges have different colors. We show that in any factorization of the complete graph Kn on n vertic...