A copy of a graph H in an edge colored graph G is called rainbow if all edges of H have distinct colors. The size anti-Ramsey number of H, denoted by ARs(H), is the smallest number of edges in a graph G such that any of its proper edge-colorings contains a rainbow copy of H. We show that ARs(Kk) = Θ(k / log k). This settles a problem of Axenovich, Knauer, Stumpp and Ueckerdt. The proof is proba...

‎an  oriented perfect path double cover (oppdc) of a‎ ‎graph $g$ is a collection of directed paths in the symmetric‎ ‎orientation $g_s$ of‎ ‎$g$ such that‎ ‎each arc‎ ‎of $g_s$ lies in exactly one of the paths and each‎ ‎vertex of $g$ appears just once as a beginning and just once as an‎ ‎end of a path‎. ‎maxov{'a} and ne{v{s}}et{v{r}}il (discrete‎ ‎math‎. ‎276 (2004) 287-294) conjectured that ...

For given graphs G and H; the Ramsey number R(G;H) is the leastnatural number n such that for every graph F of order n the followingcondition holds: either F contains G or the complement of F contains H.In this paper firstly, we determine Ramsey number for union of pathswith respect to sunflower graphs, For m ≥ 3, the sunflower graph SFmis a graph on 2m + 1 vertices obtained...

The graph Ramsey number R(G,H) is the smallest integer n such that every 2-coloring of the edges of Kn contains either a red copy of G or a blue copy of H . We find the largest star that can be removed from Kn such that the underlying graph is still forced to have a red G or a blue H . Thus, we introduce the star-avoiding Ramsey number r∗(G,H) as the smallest integer k such that every 2-colorin...

Given a family of graphs F , a graph G is F-saturated if no element of F is a subgraph of G, but for any edge e in G, some element of F is a subgraph of G + e. Let sat(n,F) denote the minimum number of edges in an F -saturated graph of order n. For graphs G,H1, . . . , Hk, we write that G → (H1, . . . , Hk) if every k-coloring of E(G) contains a monochromatic copy of Hi in color i for some i. A...

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 . We find the largest star that can be removed from Kr such that the underlying graph is still forced to have a red G or a blue H . Thus, we introduce the star-critical Ramsey number r∗(G,H) as the smallest integer k such that every 2-colorin...

The Ramsey number, r(G), of a graph G is the minimum integer N such that, in every 2-colouring of the edges of the complete graph KN on N vertices, there is a monochromatic copy of G. In 1975, Burr and Erdős posed a problem on Ramsey numbers of d-degenerate graphs, i.e., graphs in which every subgraph has a vertex of degree at most d. They conjectured that for every d there exists a constant c(...

Let G be a graph with given red-blue coloring c of the edges G. An ascending Ramsey sequence in respect to is G1, G2, …, Gk pairwise edge-disjoint subgraphs such that each subgraph Gi (1≤i≤k) monochromatic and isomorphic proper Gi+1 (1≤i≤k−1). The index ARc(G) maximum length an c. AR(G) minimum value among all colorings It shown there connection between this concept set partitions. investigated...