In this chapter G = (V,E) denotes an arbitrary undirected graph without loops, where V = {v 1, v 2,…, v n } is its vertex set and E = {e 1,e 2,…, e m } ⊂ (E ×E) is its edge set. Two edges are adjacent if they connect to a common vertex. Two vertices v i and v j are adjacent if there is an edge e = (v i ,v j ) ∈ E. Finally, if e = (v i ,v j ) ∈ E,we say e is incident to vertices v i , v j .

The Graph Coloring Problem: A Bibliographic Survey

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Maximal Sum-Free Sets of Elements of Finite Groups

We shall prove a new generalization of Vosper critical pair theorem to finite Abelian groups. We next apply this new tool to the theory of $(k,l)$-free sets in finite Abelian groups. In particular, in most cases, we describe the structure of maximal $(k,l)$-free sets and determine the maximal cardinality of such a set. This result allows us for instance to give precisions on an old result of Yap: we are able to describe completely the maximal sum-free sets with cardinality at least one third of that of the ambient group.

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A new critical pair theorem applied to sum-free sets in Abelian groups

In this paper we study sum-free subsets of the set $\{1,...,n\}$, that is, subsets of the first $n$ positive integers which contain no solution to the equation $x + y = z$. Cameron and Erdős conjectured in 1990 that the number of such sets is $O(2^{n/2})$. This conjecture was confirmed by Green and, independently, by Sapozhenko. Here we prove a refined version of their theorem, by showing that the number of sum-free subsets of $[n]$ of size $m$ is $2^{O(n/m)} {\lceil n/2 \rceil \choose m}$, for every $1 \le m \le \lceil n/2 \rceil$. For $m \ge \sqrt{n}$, this result is sharp up to the constant implicit in the $O(\cdot)$. Our proof uses a general bound on the number of independent sets of size $m$ in 3-uniform hypergraphs, proved recently by the authors, and new bounds on the number of integer partitions with small sumset.

A refinement of the Cameron-Erd\H{o}s Conjecture

Sum-free sets in vector spaces over GF(2)

Maximal Sum-Free Sets of Group Elements

Let G be a graph of order 2n+ 1 having maximum degree 2n—\\. We prove that the total chromatic number of G is 2/i if and only if e(G — w) + oc'(G — w)^n, where w is a vertex of minimum degree in G, G — w is the complement of G — w, e(G — w) is the size of G — w, and a'(G-w) is the edge independence number of G — w.

H. P. Yap

Papers

Maximal Sum-Free Sets of Group Elements

Total Chromatic Number of Graphs of Order 2n + l having Maximum Degree 2n − 1