A classical result in graph theory asserts that every graph can be oriented so that the indegree and outdegree of each vertex differ by at most 1. We study the extent to which the result generalizes to uniform hypergraphs.

subgraph. (a) x has to belong to a triangle, (b) x has to belong to a 2-clan of size at least 6, (c) x has to be connected to a vertex y such that the degree of y is at least 6, (d) x has to belong to a connected component whose

The trace algebra Cnd over a field of characteristic 0 is generated by all traces of products of d generic n × n matrices, n, d ≥ 2. Minimal sets of generators of Cnd are known for n = 2 and 3 for any d and for n = 4 and 5 and d = 2. The explicit defining relations between the generators are found for n = 2 and any d and for n = 3, d = 2 only. Defining relations of minimal degree for n = 3 and ...

It is well known that every bipartite graph with vertex classes of size n whose minimum degree is at least n/2 contains a perfect matching. We prove an analogue of this result for hypergraphs. We also prove several related results which guarantee the existence of almost perfect matchings in r-uniform hypergraphs of large minimum degree. Our bounds on the minimum degree are essentially best poss...

Using a result of Gurevich and Lewis on the word problem for finite semigroups, we give short proofs that the following theories are hereditarily undecidable: (1) finite graphs of vertex-degree at most 3; (2) finite nonvoid sets with two distinguished permutations; (3) finitedimensional vector spaces over a finite field with two distinguished endomorphisms.

--avs subject classification (1980): 05 C 65, 05 C 35; 05 B 25 Let 3f be a family of r-subsets of a finite set X. Set D(/{):max|{E: xQE€lf,}|, (maximum degree). We say that 3/f, is intersecting if for any H, H,€;( .we Itave _H ) E, #0. In this case, obuiő".rí, ottj=tcfÍh. According to a well-known conjec1ule D9r)=|ű.|l(r-|*1lr). We proveísügiítiíströnger result .Let /f, beanr-uniform, intersect...

Coherence is a positive virtue of theories, and its importance in theory construction is beyond debate. Unfortunately, however, a rigorous and computable definition of coherence is conspicuously lacking in the literature. This paper attempts to remedy this situation by formalising this notion of coherence and suggesting a measure. Roughly speaking, we suggest that a theory is coherent to the de...

A k-monocore graph is a graph which has its minimum degree and degeneracy both equal to k. Integer sequences that can be the degree sequence of some k-monocore graph are characterized as follows. A nonincreasing sequence of integers d1, . . . , dn is the degree sequence of some k-monocore graph G, 0 ≤ k ≤ n− 1, if and only if k ≤ di ≤ min {n− 1, k + n− i} and

We give a description of degree-one maps between closed, oriented 3-manifolds in terms of surgery. Namely, we show that there is a degree-one map from a closed, oriented 3-manifold M to a closed, oriented 3-manifold N if and only if M can be obtained from N by surgery about a link in N each of whose components is an unknot. We use this to interpret the existence of degree-one maps between close...

Let G be a graph of order n(G), minimum degree (G) and connectivity (G). Chartrand and Harary [Graphs with prescribed connectivities, in: P. Erdös, G. Katona (Eds.), Theory of Graphs, Academic Press, NewYork, 1968, pp. 61–63] gave the following lower bound on the connectivity (G) 2 (G)+ 2− n(G). Topp and Volkmann [Sufficient conditions for equality of connectivity and minimum degree of a graph,...