A $(k, g)$-cage is a $k$-regular graph of girth $g$ of minimum order. In this survey, we present the results of over 50 years of searches for cages. We present the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant constructions.

The linear arboncity of a graph is the minimum number of linear forests into which its lines can be decomposed. We find that the linear arboricity of every 4-regular graph is 3. This result enables us to obtain bounds for the linear arboricity of any graph in terms of its maximum degree.

A graph satisfies Axiom n if, for any sequence of 2n of its points, there is another point adjacent to the first n and not to any of the last n . We show that, for each n , all sufficiently Igrge Paley graphs satisfy Axiom n. From this we conclude a t once that several properties of graphs are not first order, including self-complementarity and regularity.A first-order theory of graphs was developed in [l]. Certain adjacency axioms were given and shown to be satisfied by almost all graphs. However, almost no specific graphs were known to satisfy them. In this note we exhibit such graphs and draw some conclusions concerning first-order sentences.The graph-theoretic notation and terminology of [7] is used.We begin with a summary of the first-order theory of graphs. The firstorder language L consists of predicate symbols for equality and adjacency, propositional connectives not, and, and or, the existential quanti$er, and the propositional constants true and false. The axioms of the theory include Axiom 0, which stipulates that a graph has at least two points and that adjacency is an irreflexive, symmetric relation. For n 2 1, Axiom n is satisfied by a graph if, for any sequence of 2n of its points, there is another point adjacent to the first n points but not to the last n. A useful result from[l] is now stated.Theorem A. For every L-sentence 8, either 8 is deducible from finitely many of these axioms, or "not 8" is so deducible. I Our present purpose is to exhibit a family of graphs satisfying Axiom n, for any given n. Ifp is a prime congruent to 1 modulo 4, then the Paley graph

We consider a variation on the problem of determining the chromatic number of the Euclidean plane and define the ε-unit distance graph to be the graph whose vertices are the points of E 2 , in which two points are adjacent whenever their distance is within ε of 1. For certain values of ε we are able to show that the chromatic number is exactly 7. For some smaller values we show the chromatic number is at least 5. We offer a conjecture, with some supporting evidence, that for any ε > 0 the chromatic number is 7.

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This paper studies lower bounds for classical multicolor Ramsey numbers, first by giving a short overview of past results, and then by presenting several general constructions establishing new lower bounds for many diagonal and off-diagonal multicolor Ramsey numbers. In particular, we improve several lower bounds for $R_k(4)$ and $R_k(5)$ for some small $k$, including $415 \le R_3(5)$, $634 \le R_4(4)$, $2721 \le R_4(5)$, $3416 \le R_5(4)$ and $26082 \le R_5(5)$. Most of the new lower bounds are consequences of general constructions.

a b s t r a c tBertrand, Charon, Hudry and Lobstein studied, in their paper in 2004 [1], r-locatingdominating codes in paths P n . They conjectured that if r ≥ 2 is a fixed integer, then the smallest cardinality of an r-locating-dominating code in P n , denoted by M LD r (P n ), satisfies M LD r (P n ) = ⌈(n + 1)/3⌉ for infinitely many values of n. We prove that this conjecture holds. In fact, we show a stronger result saying that for any r ≥ 3 we have M LD r (P n ) = ⌈(n + 1)/3⌉ for all n ≥ n r when n r is large enough. In addition, we solve a conjecture on location-domination with segments of even length in the infinite path.

We present an alternate proof of the fact that given any 4-coloring of the plane there exist two points unit distance apart which are identically colored.