Search

This paper deals with extremal problems concerning colorings of partial Steiner systems. We establish a new sufficient condition for r-colorability of a hypergraph from some class of such systems in terms of maximum vertex degree. Moreover, as a corollary we obtain a new lower bound for the threshold probability for r-colorability of a random hypergraph in a binomial model. [ABSTRACT FROM AUTHOR]

Is it proves that every connected graph with s vertices of degree 1 and 3 anf t vertices of degree at least 4 has a spanning tree with at least $ \frac{1}{3}t + \frac{1}{4}s + \frac{3}{2} $ leaves. Infinite series of grpahs showing that our bound is tight are given. [ABSTRACT FROM AUTHOR]

Let G be a graph that can be drawn on a plane in such a way that any edge intersects at most one other edge. It is proved that the chromatic number of G does not exceed 7. The bound $ \chi (G)\leq \frac{{9+\sqrt{17+64g }}}{2} $ is also proved for a graph G that can be drawn on a surface of genus g ( g ≥ 1) in such a way that any edge intersects at most one other edge. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

Let G be a bipartite graph without loops and multiple edges on v ≥ 4 vertices, which can be drawn on a plane in such a way that any edge intersects at most one other edge. It is proved that such a graph has at most 3 v - 8 edges for even v ≠ 6 and at most 3 v - 9 edges for odd v and for v = 6. For all v ≥ 4, examples showing that these bounds are tight are constructed. At the end of the paper, it is discussed a question about drawings of complete bipartite graphs on a plane such that any edge intersects at most one other edge. Bibliography: 6 titles. [ABSTRACT FROM AUTHOR]

It is proved that every connected graph with s vertices of degree 3 and t vertices of degree at least 4 has a spanning tree with $ \frac{2}{5}t+\frac{1}{5}s+\alpha $ leaves, where $ \alpha \geq \frac{8}{5} $. Moreover, α ≥ 2 for all graphs, except for three exclusions. All exclusions are regular graphs of degree 4, and they are explicitly described in the paper. An infinite series of graphs containing only vertices of degree 3 and 4, for which the maximal number of leaves in a spanning tree is equal to $ \frac{2}{5}t+\frac{1}{5}s+2 $, is presented. Therefore it is proved that the bound is tight. Bibliography: 12 titles. [ABSTRACT FROM AUTHOR]

In this paper, we present lower bounds for the number of vertices in a graph with a large chromatic number that does not contain small odd cycles. Bibliographyy 6 titles. [ABSTRACT FROM AUTHOR]

The Q-index of a simple graph is the largest eigenvalue of its signless Laplacian, or Q-matrix. In our previous paper [] we gave three lower and three upper bounds for the Q-index of nested split graphs, also known as threshold graphs. In this paper, we give another two upper bounds, which are expressed as solutions of cubic equations (in contrast to quadratics from []). Some computational results are also included. [ABSTRACT FROM AUTHOR]