A path in an edge-colored graph is called a monochromatic if all edges of the have same color. We call k paths \(P_1,\ldots ,P_k\) rainbow every \(P_i\) and for any two \(i\ne j\), \(P_j\) different colors. An edge-coloring G said to be k-edge-connection coloring (or \(RMC_k\)-coloring short) distinct vertices are connected by at least paths. use \(rmc_k(G)\) denote maximum number colors that e...

In this paper, a combination of algorithms for graph coloring is discussed. The algorithms of KargerMotwani-Sudan and Blum-Karger can be combined in a simple way to yield a polynomial-time algorithm for an Õ(n7/18)-coloring of any n-vertex 4-colorable graph. This result can be generalized to k-colorable graphs to obtain a coloring with Õ(n 1− 1 (k+1)/3−4/(11k2−11k) ) colors, which slightly impr...

In 1967 Harary, Hedetniemi, and Prins showed that every graph G admits a complete t-coloring for t with χ(G)≤t≤ψ(G), where χ(G) denotes the chromatic number of ψ(G) achromatic which is maximum r r-coloring. Recently, Edwards Rza̧żewski (2020) this result fails hypergraphs by proving integer k k≥9, there exists k-uniform hypergraph H χ(H)-coloring ψ(H)-coloring, but no some χ(H)<t<ψ(H). They also...

A rainbow coloring of a graph is a coloring of the edges with distinct colors. We prove the following extension of Wilson’s Theorem. For every integer k there exists an n0 = n0(k) so that for all n > n0, if n mod k(k − 1) ∈ {1, k}, then every properly edge-colored Kn contains (n 2 ) / (k 2 ) pairwise edge-disjoint rainbow copies of Kk. Our proof uses, as a main ingredient, a double application ...

A 2-hued coloring of a graph G (also known as conditional (k, 2)-coloring and dynamic coloring) is a coloring such that for every vertex v ∈ V (G) of degree at least 2, the neighbors of v receive at least 2 colors. The smallest integer k such that G has a 2-hued coloring with k colors, is called the 2-hued chromatic number of G and denoted by χ2(G). In this paper, we will show that if G is a re...

Abstrac t . Graph coloring is the dominant paradigm for global register allocation [8, 7, 4]. Coloring allocators use an interference graph, Z, to model conflicts that prevent two values from sharing a register. Nodes in 2: represent live ranges. An edge between two nodes indicates that they are simultaneously live azld, thus, cannot share a register. The allocator tries to construct a k-colori...

Given a hypergraph H = (V, E), a coloring of its vertices is said to be conflict-free if for every hyperedge S ∈ E there is at least one vertex in S whose color is distinct from the colors of all other vertices in S. More generally if every hyperedge has k-distinct color then the coloring is called k-strong conflict-free coloring. In this paper, we present a polynomial k-strong conflict-free co...

Given an integer k ≥ 2, we consider vertex colorings of graphs in which no k-star subgraph Sk = K1,k is polychromatic. Equivalently, in a star-[k]-coloring the closed neighborhood N[v] of each vertex v can have at most k different colors on its vertices. The maximum number of colors that can be used in a star-[k]-coloring of graph G is denoted by χ̄k⋆(G) and is termed the star-[k] upper chromati...

This thesis studies both several extremal problems about coloring of graphs and a labeling problem on graphs. We consider colorings of graphs that are either embeddable in the plane or have low maximum degree. We consider three problems: coloring the vertices of a graph so that no adjacent vertices receive the same color, coloring the edges of a graph so that no adjacent edges receive the same ...

A k-total-coloring of a graph G is a coloring of vertex set and edge set using k colors such that no two adjacent or incident elements receive the same color. In this paper, we prove that if G is a planar graph with maximum ∆ ≥ 8 and every 6-cycle of G contains at most one chord or any chordal 6-cycles are not adjacent, then G has a (∆ + 1)-total-coloring.