Primal graphs with small degrees

Edge coloring of graphs with small average degrees

Let G be a simple graph with average degree . d and maximum degree . It is proved, in this paper that G is not critical if . d6 6 and ¿ 8, or . d6 20 3 and ¿ 9. This result generalizes earlier results of Vizing (Metody Diskret. Analiz. 5 (1965) 9), Mel’nikov (Mat. Zametki 7 (1970) 671) and Hind and Zhao (Discrete Math. 190 (1998) 107) and Yan and Zhao (Graphs Combin. 16 (2) (2000) 245). It also...

Graphs with unavoidable subgraphs with large degrees

Let `,(n, m) denote the class of simple graphs on n vertices and m edges and let G C `6(n, m) . There are many results in graph theory giving conditions under which G contains certain types of subgraphs, such as cycles of given lengths, complete graphs, etc . For example, Turan's theorem gives a sufficient condition for G to contain a K,,„ in terms of the number of edges in G . In this paper we...

Small graphs with exactly two non-negative eigenvalues

Let $G$ be a graph with eigenvalues $lambda_1(G)geqcdotsgeqlambda_n(G)$. In this paper we find all simple graphs $G$ such that $G$ has at most twelve vertices and $G$ has exactly two non-negative eigenvalues. In other words we find all graphs $G$ on $n$ vertices such that $nleq12$ and $lambda_1(G)geq0$, $lambda_2(G)geq0$ and $lambda_3(G)0$, $lambda_2(G)>0$ and $lambda_3(G)

Graphs with degrees from prescribed intervals

List homomorphisms of graphs with bounded degrees

In a series of papers we have classified the complexity of list homomorphism problems. Here we investigate the effect of restricting the degrees of the input graphs. It turns out that the complexity does not change (except when the degree bound is two). We obtain similar results on restricting the size of the lists. We contrast these results with facts about some variants of the list homomorphi...