A path in an edge-colored graph G, where adjacent edges may be colored the same, is called a rainbow path if no two edges of G are colored the same. For a κ-connected graph G and an integer k with 1 ≤ k ≤ κ, the rainbow k-connectivity rck(G) of G is defined as the minimum integer j for which there exists a j-edgecoloring of G such that every two distinct vertices of G are connected by k interna...

A graph is called integral if all the eigenvalues of its adjacency matrix are integers. In this paper, we give a useful sufficient and necessary condition for complete r-partite graphs to be integral, from which we can construct infinite many new classes of such integral graphs. It is proved that the problem of finding such integral graphs is equivalent to the problem of solving some Diophantin...

Erdős, Rubin, and Taylor found a nice correspondence between the minimum order of a complete bipartite graph that is not r-choosable and the minimum number of edges in an r-uniform hypergraph that is not 2-colorable (in the ordinary sense). In this note we use their ideas to derive similar correspondences for complete kpartite graphs and complete k-uniform k-partite hypergraphs.

The tree partition number of an r-edge-colored graph G, denoted by tr(G), is the minimum number k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex-disjoint monochromatic trees. We determine t2(K(n1, n2, . . . , nk)) of the complete k-partite graph K(n1, n2, . . . , nk). In particular, we prove that t2(K(n,m)) = (m − 2)/2n + 2, whe...

In 1987, Kolaitis, Prömel and Rothschild proved that, for every fixed r ∈ N, almost every n-vertex Kr+1-free graph is r-partite. In this paper we extend this result to all functions r = r(n) with r 6 (log n). The proof combines a new (close to sharp) supersaturation version of the Erdős–Simonovits stability theorem, the hypergraph container method, and a counting technique developed by Balogh, ...

For every real p > 0 and simple graph G, set f (p,G) = ∑ u∈V (G) d (u) , and let φ (r, p, n) be the maximum of f (p,G) taken over all Kr+1-free graphs G of order n. We prove that, if 0 < p < r, then φ (r, p, n) = f (p, Tr (n)) , where Tr (n) is the r-partite Turan graph of order n. For every p ≥ r + ⌈√ 2r ⌉ and n large, we show that φ (p, n, r) > (1 + ε) f (p, Tr (n)) for some ε = ε (r) > 0. Ou...

Let f(n, r, k) be the minimal number such that every hypergraph larger than f(n, r, k) contained in ([n] r ) contains a matching of size k, and let g(n, r, k) be the minimal number such that every hypergraph larger than g(n, r, k) contained in the r-partite r-graph [n]r contains a matching of size k. The Erdős-Ko-Rado theorem states that f(n, r, 2) = (n−1 r−1 ) (r ≤ n 2 ) and it is easy to show...

Let R be a ring with unity. The graph Γ(R) is a graph with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if Ra + Rb = R. Let Γ2(R) is the subgraph of Γ(R) induced by the non-unit elements. H.R. Maimani et al. [H.R. Maimani et al., Comaximal graph of commutative rings, J. Algebra 319 (2008) 1801-1808] proved that: “If R is a commutative ring with unity a...

the chromatic number of a graph g, denoted by χ(g), is the minimum number of colors such that g can be colored with these colors in such a way that no two adjacent vertices have the same color. a clique in a graph is a set of mutually adjacent vertices. the maximum size of a clique in a graph g is called the clique number of g. the turán graph tn(k) is a complete k-partite graph whose partition...

Suppose a graph G have n vertices, m edges, and t triangles. Letting λn (G) be the largest eigenvalue of the Laplacian of G and μn (G) be the smallest eigenvalue of its adjacency matrix, we prove that λn (G) ≥ 2m2 − 3nt m (n2 − 2m) , μn (G) ≤ 3n3t− 4m3 nm (n2 − 2m) , with equality if and only if G is a regular complete multipartite graph. Moreover, if G is Kr+1-free, then λn (G) ≥ 2mn (r − 1) (...