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The graph Ramsey number R(G,H) is the smallest integer r such that every 2-coloring of the edges of Kr contains either a red copy of G or a blue copy of H. The star-critical Ramsey number r∗(G,H) is the smallest integer k such that every 2-coloring of the edges of Kr − K1,r−1−k contains either a red copy of G or a blue copy of H. We will classify the critical graphs, 2-colorings of the complete graph on R(G,H) − 1 vertices with no red G or blue H, for the path-path Ramsey number. This classification will be used in the proof of r∗(Pn, Pm).

In 1968, Vizing conjectured that for any edge chromatic critical graph G = (V,E) with maximum degree ∆ and independence number α(G), α(G) ≤ |V | 2 . It is known that α(G) < 3∆−2 5∆−2 |V |. In this paper we improve this bound when ∆ ≥ 4. Our precise result depends on the number n2 of 2-vertices in G, but in particular we prove that α(G) ≤ 3∆−3 5∆−3 |V | when ∆ ≥ 5 and n2 ≤ 2(∆− 1).

The oriented chromatic number of an oriented graph $\vec G$ is the minimum order of an oriented graph $\vev H$ such that $\vec G$ admits a homomorphism to $\vev H$. The oriented chromatic number of an undirected graph $G$ is then the greatest oriented chromatic number of its orientations. In this paper, we introduce the new notion of the upper oriented chromatic number of an undirected graph $G$, defined as the minimum order of an oriented graph $\vev U$ such that every orientation $\vec G$ of $G$ admits a homomorphism to $\vec U$. We give some properties of this parameter, derive some general upper bounds on the ordinary and upper oriented chromatic numbers of Cartesian, strong, direct and lexicographic products of graphs, and consider the particular case of products of paths., 14 pages

For a graph G = (V,E) without isolated vertices, a subset D of vertices of V is a total dominating set (TDS) of G if every vertex in V is adjacent to a vertex in D. The total domination number t(G) is the minimum cardinality of a TDS of G. A subset D of V which is a total dominating set, is a locating-total dominating set, or just a LTDS of G, if for any two distinct vertices u and v of V (G) n D, NG(u)\ D 6 NG(v)\ D. The locating-total domination number t L(G)

In a graph, by deflnition, the weight of a (proper) coloring with positive integers is the sum of the colors. The chromatic sum is the minimum weight, taken over all the proper colorings. The minimum number of colors in a coloring of minimum weight is the cost chromatic number or strength of the graph. We derive general upper bounds for the strength, in terms of a new parameter of representations by edge intersections of hypergraphs.

Hadwiger's Conjecture seems difficult to attack, even in the very special case of graphs G of independence number α (G) = 2. We present some results in this special case.