A locally irregular graph is a in which the end vertices of every edge have distinct degrees. coloring G any such that each colors induces subgraph G. colorable if it allows coloring. The chromatic index G, denoted by χirr′(G), smallest number used local irregularity conjecture claims all graphs, except odd-length paths, cycles and certain class cacti are three colors. As valid for graphs with ...

An edge-coloring of a graph G with colors 1, . . . , t is an interval t-coloring if all colors are used, and the colors of edges incident to each vertex of G are distinct and form an interval of integers. A graph G is interval colorable if it has an interval t-coloring for some positive integer t. In this note we prove that K1,m,n is interval colorable if and only if gcd(m+ 1, n+ 1) = 1, where ...

Fix positive integers k′, d′, k, d such that k′/d′ > k/d ≥ 2. If P is a set of vertices in a (k, d)-colorable graph G, and any two vertices of P are separated by distance at least 2 ⌈ kk′ 2(k′d−kd′) ⌉ , then every coloring of P with colors in Zk′ extends to a (k′, d′)coloring of G. If k′d − kd′ = 1 and $k′/d′% = $k/d%, then this distance threshold is nearly sharp. The proof of this includes sho...

A clique-coloring of a given graph G is a coloring of the vertices of G such that no maximal clique of size at least two is monocolored. The clique-chromatic number of G is the least number of colors for which G admits a clique-coloring. It has been proved that every planar graph is 3-clique colorable and every claw-free planar graph, different from an odd cycle, is 2-clique colorable. In this ...

A graph G is strongly set colorable if V (G) ∪ E(G) can be assigned distinct nonempty subsets of a set of order n, where |V (G)| + |E(G)| = 2n − 1, such that each edge is assigned the symmetric difference of its end vertices. The principal result is that the path P2n−1 is strongly set colorable for n ≥ 5, disproving a conjecture of S.M. Hegde. We also prove another conjecture of Hegde on a rela...

A mixed hypergraph consists of two families of subsets of the vertex set: the V-edges and the C-edges. In a suitable coloring of a mixed hypergraph, every C-edge has at least two vertices of the same color, and every V-edge has at least two vertices colored differently. The largest and smallest possible numbers of colors in a coloring are called the upper and lower chromatic numbers, X and X, r...