Given a simple graph G, a harmonious coloring of G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number is the least integer k for which G admits a harmonious coloring with k colors. Extending previous NP-completeness results of the harmonious coloring problem on subclasses of chordal and co-chordal graphs, we prove that...

For a fixed simple digraph F and given D , an - free k -coloring of is vertex-coloring in which no induced copy monochromatic. We study the complexity deciding for whether admits -free -coloring. Our main focus on restriction problem to planar input digraphs, where it only interesting cases ? { 2 3 } . From known results follows that every whose underlying graph not forest, 2-coloring, with ? (...

An r-edge coloring of a graph G is a mapping h : E(G) → [r], where h(e) is the color assigned to edge e ∈ E(G). An exact r-edge coloring is an r-edge coloring h such that there exists an e ∈ E(G) with h(e) = i for all i ∈ [r]. Let h be an edge coloring of G. We say G is rainbow if no two edges in G are assigned the same color by h. The anti-Ramsey number, AR(G,n), is the smallest integer r such...

Proper vertex coloring c of a graph G is graceful if k-coloring for k∈{1,2,3,…}. Definition G=(V,E) proper c:V(G)→{1,2,…,k);k≥2, which induces edge c':E(G)→{1,2,…,k-1} defined c'(uv)=|c(u)-c(v)|. The minimum from can be colored with called chromatic number notation χg (G). In this paper, we will investigate the amalgamation tree family some path graph, centipede broom and E graph.

In this paper, we prove that the harmonious coloring problem is NP-complete for connected interval and permutation graphs. Given a simple graph G, a harmonious coloring of G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number is the least integer k for which G admits a harmonious coloring with k colors. Extending previo...

We consider proper online colorings of hypergraphs defined by geometric regions. Among others we prove that there is an online coloring method that colors N intervals on the line using Θ(logN/k) colors such that for every point p, contained in at least k intervals, there exist two intervals containing p and having different colors. We prove the corresponding (but not equivalent) result about on...

For 1≤s1≤s2≤…≤sk and a graph G, packing (s1,s2,…,sk)-coloring of G is partition V(G) into sets V1,V2,…,Vk such that, for each 1≤i≤k the distance between any two distinct x,y∈Vi at least si+1. The chromatic number, χp(G), smallest k that has (1,2,…,k)-coloring. It known there are trees maximum degree 4 subcubic graphs with arbitrarily large χp(G). Recently, was series papers on (s1,s2,…,sk)-colo...

A variation of graph coloring known as a t-tone k-coloring assigns a set of t colors to each vertex of a graph from the set {1, . . . , k}, where the sets of colors assigned to any two vertices distance d apart share fewer than d colors in common. The minimum integer k such that a graph G has a ttone k-coloring is known as the t-tone chromatic number. We study the 2-tone chromatic number in thr...

Consider edge colorings of directed graphs where edges of the form v1v2 and v2v3 must have different colors. Here, v1 ≠ v2 , v2 ≠ v3 but v1 = v3 is possible. It is known that this coloring induces a vertex coloring by sets of edge colors, in which edge v1v2 in the graph implies that the set color of v1 contains an element not in the set color of v2; conversely, each such set coloring of vertice...

Previously in class we proved that if we color the edges of K6 using blue or red color, then either there is a blue K3 or a red K3 as a subgraph. Here Kn is the complete graph (every pair of vertices are connected) on n vertices. We can generalize the above concept and ask, are there complete graphs for which any 2-coloring gives rise to either a blue Kk or a red Kl. It has been shown that ther...