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We study pairs of graphs (H_1,H_2) such that every graph with the densities of H_1 and H_2 close to the densities of H_1 and H_2 in a random graph is quasirandom; such pairs (H_1,H_2) are called forcing. Non-bipartite forcing pairs were first discovered by Conlon, Han, Person and Schacht [Weak quasi-randomness for uniform hypergraphs, Random Structures Algorithms 40 (2012), 1-38]: they showed that (K_t,F) is forcing where F is the graph that arises from K_t by iteratively doubling its vertices and edges in a prescribed way t times. Reiher and Schacht [Forcing quasirandomness with triangles, Forum of Mathematics, Sigma 7, 2019] strengthened this result for t=3 by proving that two doublings suffice and asked for the minimum number of doublings needed for t>3. We show that (t+2)/2 doublings always suffice.
A graph H is k-common if the number of monochromatic copies of H in a k-edge-coloring of K_n is asymptotically minimized by a random coloring. For every k, we construct a connected non-bipartite k-common graph. This resolves a problem raised by Jagge
In this paper, we classify the connected non-bipartite integral graphs with spectral radius three.
The global forcing number of a graph G is the minimal cardinality of an edge subset discriminating all perfect matchings of G, denoted by gf(G). For any perfect matching M of G, the minimal cardinality of an edge subset S in E(G)-M such that G-S has
Zero forcing is a combinatorial game played on a graph with a goal of turning all of the vertices of the graph black while having to use as few unforced moves as possible. This leads to a parameter known as the zero forcing number which can be used t
Let $G=(V,E)$ be a finite connected graph along with a coloring of the vertices of $G$ using the colors in a given set $X$. In this paper, we introduce multi-color forcing, a generalization of zero-forcing on graphs, and give conditions in which the