A λ-coloring of a graph G is an assignment of colors from the integer set {0, . . . , λ} to the vertices of the graph G such that vertices at distance of at most two get different colors and adjacent vertices get colors which are at least two apart. The problem of finding λ-colorings with optimal or near-optimal λ arises in the context of radio frequency assignment. We show that the problem of ...

Eroh and Oellerman defined BRR(G1, G2) as the smallest N such that any edge coloring of the complete bipartite graph KN,N contains either a monochromatic G1 or a multicolored G2. We restate the problem of determining BRR(K1,λ,Kr,s) in matrix form and prove estimates and exact values for several choices of the parameters. Our general bound uses Füredi’s result on fractional matchings of uniform ...

An(H , H)-good coloring is the coloring of the edges of a (hyper)graph H such that no subgraph H ⊆ H is monochrome or rainbow. Similarly, we define an (H , H)-proper coloring to be the coloring of the vertices of H with forbidden monochromatic and rainbow copies of H . An (H , Kt)-good coloring is also known as a mixed Ramsey coloring when H = Kn is a complete graph, and an (H , Kt)-proper colo...

f : ∪A → ρ is called a conflict free coloring of the set-system A (with ρ colors) if ∀A ∈ A ∃ ζ < ρ ( |A ∩ f−1{ζ}| = 1 ). The conflict free chromatic number χCF(A) of A is the smallest ρ for which A admits a conflict free coloring with ρ colors. A is a (λ, κ, μ)-system if |A| = λ, |A| = κ for all A ∈ A, and A is μ-almost disjoint, i.e. |A ∩ A′| < μ for distinct A,A′ ∈ A. Our aim here is to stud...

Motivated by the definition of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology, and the framework through which it was studied, we introduce the colinear coloring on graphs. We provide an upper bound for the chromatic number χ(G), for any graph G, and show that G can be colinearly colored in polynomial time by proposing a simple algorithm. The c...

Let G be a simple graph with n vertices. The coloring complex ∆(G) was defined by Steingrı́msson, and the homology of ∆(G) was shown to be nonzero only in dimension n − 3 by Jonsson. Hanlon recently showed that the Eulerian idempotents provide a decomposition of the homology group Hn−3(∆(G)) where the dimension of the j component in the decomposition, H n−3(∆(G)), equals the absolute value of th...

Let G be a simple graph with n vertices. The coloring complex ∆(G) was defined by Steingrı́msson, and the homology of ∆(G) was shown to be nonzero only in dimension n − 3 by Jonsson. Hanlon recently showed that the Eulerian idempotents provide a decomposition of the homology group Hn−3(∆(G)) where the dimension of the j component in the decomposition, H n−3(∆(G)), equals the absolute value of th...

Motivated by the definition of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology [9], and the framework through which it was studied, we introduce the linear coloring on graphs. We provide an upper bound for the chromatic number χ(G), for any graph G, and show that G can be linearly colored in polynomial time by proposing a simple linear coloring ...