Yared Nigussie

On a new reformulation of Hadwiger's conjecture

Assuming that every proper minor closed class of graphs contains a maximum with respect to the homomorphism order, we prove that such a maximum must be homomorphically equivalent to a complete graph. This proves that Hadwigers conjecture is equivalent to saying that every minor closed class of graphs contains a maximum with respect to homomorphism order. Let F be a finite set of 2-connected graphs, and let C be the class of graphs with no minor from F. We prove that if C has a maximum, then any maximum of C must be homomorphically equivalent to a complete graph. This is a special case of a conjecture of Nesetril and Ossona de Mendez.

Maximal independent sets in minimum colorings

Every graph G contains a minimum vertex-coloring with the property that at least one color class of the coloring is a maximal independent set (equivalently, a dominating set) in G. Among all such minimum vertex-colorings of the vertices of G, a coloring with the maximum number of color classes that are dominating sets in G is called a dominating-@g-coloring of G. The number of color classes that are dominating sets in a dominating-@g-coloring of G is defined to be the dominating-@g-color number of G. In this paper, we continue to investigate the dominating-@g-color number of a graph first defined and studied in [1].

Algorithm for finding structures and obstructions of tree ideals

Let I be any topological minor closed class of trees (a tree ideal). A classical theorem of Kruskal [Well-quasi-ordering, the Tree Theorem, and Vazsonyis conjecture, Trans. Am. Math. Soc. 95 (1960) 210-223] states that the set O(I) of minimal non-members of I is finite. On the other hand, a finite structural description S(I) is developed by Robertson, et al. [Structural descriptions of lower ideals of trees, Contemp. Math. 147 (1993) 525-538]. Given either of the two finite characterizations of I, we present an algorithm that computes the other.

Finite dualities and map-critical graphs on a fixed surface

Let K be a class of graphs. A pair (F,U) is a finite duality in K if U@?K, F is a finite set of graphs, and for any graph G in K we have G=

Short proofs for two theorems of Chien, Hell and Zhu

In (J Graph Theory 33 (2000), 14–24), Hell and Zhu proved that if a series–parallel graph G has girth at least 2⌊(3k−1)/2⌋, then χc(G)≤4k/(2k−1). In (J Graph Theory 33 (2000), 185–198), Chien and Zhu proved that the girth condition given in (J Graph Theory 33 (2000), 14–24) is sharp. Short proofs of both results are given in this note.

Extended Gallai's Theorem

Abstract Let G and H be graphs. We say G is H-critical, if every proper subgraph of G except G itself is homomorphic to H. This generalizes the widely known concept of k-color-critical graphs, as they are the case H = K k − 1 . In 1963 [T. Gallai, Kritiche Graphen, I., Magyar Tud. Akad. Mat. Kutato Int. Kozl. 8 (1963), 373-395], Gallai proved that the vertices of degree k in a K k -critical graph induce a subgraph whose blocks are either odd cycles or complete graphs. We generalize Gallais Theorem for every H-critical graph, where H = K k − 2 + H ′ , (the join of a complete graph K k − 2 with any graph H ′ ). This answers one of the two unknown cases of a problem given in [J. Nesetřil, Y. Nigussie, Finite dualities and map-critical graphs on a fixed surface. (Submitted to Journal of Combin. Theory, Series B)]. We also propose an open question, which may be a characterization of all graphs for which Gallais Theorem holds.

Finite duality for some minor closed classes

Abstract Let K be a class of finite graphs and F = { F 1 , F 2 , … , F m } be a set of finite graphs. Then, K is said to have finite-duality if there exists a graph U in K such that for any graph G in K , G is homomorphic to U if and only if F i is not homomorphic to G, for all i = 1 , 2 , … , m . Nesetřil asked in [J. Nesetřil, Homonolo Combinatorics Workshop, Nova Louka, Czech Rep., (2003)] if non-trivial examples can be found. In this note, we answer this positively by showing classes containing arbitrary long anti-chains and yet having the finite-duality property.