Given a graph H, a graph G is called a Ramsey graph of H if there is a monochromatic copy of H in every coloring of the edges of G with two colors. [565][566][567][568][569][570][571][572][573][574][575][576][577][578][579][580][581][582] are not Ramsey equivalent. These are the only structural graph parameters we know that "distinguish" two graphs in the above sense. This paper provides further supportive evidence for a negative answer to the question of Fox et al. by claiming that for wide classes of graphs, chromatic number is a distinguishing parameter. In addition, it is shown here that all stars and paths and all connected graphs on at most 5 vertices are not Ramsey equivalent to any other connected graph. Moreover two connected graphs are not Ramsey equivalent if they belong to a special class of trees or to classes of graphs with clique-reduction properties.

It is well-known that the graphs not containing a given graph H as a subgraph have bounded chromatic number if and only if H is acyclic. Here we consider ordered graphs, i.e., graphs with a linear ordering ≺ on their vertex set, and the function

We define the induced arboricity of a graph G, denoted by ia(G), as the smallest k such that the edges of G can be covered with k induced forests in G. This notion generalizes the classical notions of the arboricity and strong chromatic index.For a class F of graphs and a graph parameter p, let p(F) = sup{p(G) | G ∈ F}. We show that ia(F) is bounded from above by an absolute constant depending only on F, that is ia(F) = ∞ if and only if χ(F∇ 1 /2) = ∞, where F∇ 1 /2 is the class of 1 /2-shallow minors of graphs from F and χ is the chromatic number.Further, we give bounds on ia(F) when F is the class of planar graphs, the class of d-degenerate graphs, or the class of graphs having tree-width at most d. Specifically, we show that if F is the class of planar graphs, then 8 ≤ ia(F) ≤ 10.In addition, we establish similar results for so-called weak induced arboricities and star arboricities of classes of graphs.

a b s t r a c tA vertex-coloring of a hypergraph is conflict-free, if each edge contains a vertex whose color is not repeated on any other vertex of that edge. Let f (r, ∆) be the smallest integer k such that each r-uniform hypergraph of maximum vertex degree ∆ has a conflict-free coloring with at most k colors. As shown by Pach and Tardos, similarly to a classical Brooks' type theorem for hypergraphs, f (r, ∆) ≤ ∆ + 1. Compared to Brooks' theorem, according to which there is only a couple of graphs/hypergraphs that attain the ∆ + 1 bound, we show that there are several infinite classes of uniform hypergraphs for which the upper bound is attained. We provide bounds on f (r, ∆) in terms of ∆ for large ∆ and establish the connection between conflict-free colorings and so-called {t, r − t}-factors in r-regular graphs. Here, a {t, r − t}-factor is a factor in which each degree is either t or r − t.Among others, we disprove a conjecture of Akbari and Kano (2014) stating that there is a {t, r − t}-factor in every r-regular graph for odd r and any odd t < r 3 .

The induced arboricity of a graph G is the smallest number of induced forests covering the edges of G. This is a well-defined parameter bounded from above by the number of edges of G when each forest in a cover consists of exactly one edge.