Improved bounds on linear coloring of plane graphs

Linear coloring of planar graphs with large girth

Let G = (V, E) be a connected and simple graphs with vertex set V and edge set E. A coloring of graph G is rainbow connected if there is a rainbow path that connects each two vertices of graph G. The minimum k such that G has a rainbow-connected using k colors of the edges of G is the rainbow connection number rc(G) of G. A graph with a bijective mapping f : E → {1, 2, …, |E|}. The sums of each paired vertex has distinct value, defined as ∑ e ∈E(v)f(e). Thus, the function of G clearly an antimagic labeling if the sums of each paired vertex has distinct value. It is clear that rainbow antimagic connection number is the smallest number of colors which are needed to make G rainbow connected, denoted by rcA (G). A bijection function f : E → {1, 2, …, |E|} is called a rainbow antimagic labeling if there is a rainbow path between every pair of vertices and for each edge e = uv ∈ E(G), the weight w(e) = f(u) + f(v). A graph G is rainbow antimagic if G has a rainbow antimagic labeling. In this paper, we will analyze the rainbow antimagic coloring of related book graph.

Symmetry, such as structural symmetry, color symmetry and so on, plays an important role in graph coloring. In this paper, we use structural symmetry and color symmetry to study the characterization for the neighbor-distinguishing index of planar graphs. Let G be a simple graph with no isolated edges. The neighbor-distinguishing edge coloring of G is a proper edge coloring of G such that any two adjacent vertices admit different sets consisting of the colors of their incident edges. The neighbor-distinguishing index χa′(G) of G is the smallest number of colors in such an edge coloring of G. It was conjectured that if G is a connected graph with at least three vertices and G≠C5, then χa′(G)≤Δ+2. In this paper, we show that if G is a planar graph with maximum degree Δ≥13, then Δ≤χa′(G)≤Δ+1, and, further, χa′(G)=Δ+1 if and only if G contains two adjacent vertices of maximum degree.

A proper coloring of a graph is a labeled partition of its vertices into parts which are independent sets. In this paper, given a positive integer j and a family ℱ of connected graphs, we consider proper colorings in which we require that the union of any j color classes induces a subgraph which has no copy of any member of ℱ. This generalizes some well-known types of proper colorings like acyclic colorings (where j = 2 and ℱis the collection of all even cycles) and star colorings (where j = 2 and ℱconsists of just a path on 4 vertices), etc. For this type of coloring, we obtain an upper bound of O(d(k − 1)/(k − j)) on the minimum number of colors sufficient to obtain such a coloring. Here, d refers to the maximum degree of the graph and k is the size of the smallest member of ℱ. For the case of j = 2, we also obtain lower bounds on the minimum number of colors needed in the worst case. As a corollary, we obtain bounds on the minimum number of colors sufficient to obtain proper colorings in which the union of any j color classes is a graph of bounded treewidth. In particular, using O(d8/7) colors, one can obtain a proper coloring of the vertices of a graph so that the union of any two color classes has treewidth at most 2. We also show that this bound is tight within a multiplicative factor of O((logd)1/7). We also consider generalizations where we require simultaneously for several pairs (ji, ℱi) (i = 1, …, l) that the union of any ji color classes has no copy of any member of ℱi and obtain upper bounds on the corresponding chromatic numbers. © 2011 Wiley Periodicals, Inc. J Graph Theory 66: 213–234, 2011 © 2011 Wiley Periodicals, Inc.

Proper conflict-free list-coloring, odd minors, subdivisions, and layered treewidth

For any graph $G$, the $k$-improper chromatic number $χ ^k(G)$ is the smallest number of colours used in a colouring of $G$ such that each colour class induces a subgraph of maximum degree $k$. We investigate the ratio of the $k$-improper chromatic number to the clique number for unit disk graphs and random unit disk graphs to extend results of [McRe99, McD03] (where they considered only proper colouring).

In this paper we consider a colouring version of the general position problem. The gp-chromatic number is the smallest number of colours needed to colour V(G) such that each colour class has the no-three-in-line property. We determine bounds on this colouring number in terms of the diameter, general position number, size, chromatic number, cochromatic number and total domination number and prove realisation results. We also determine it for several graph classes, including Kneser graphs K(n,2), line graphs of complete graphs, complete multipartite graphs, block graphs and Cartesian products. We also show that the gp-colouring problem is NP-complete.

On some domination colorings of graphs

Let G be a connected and undirected graph. Vertex coloring in a graph G is a mapping from the set of vertices in G to the set of colors such that every two adjacent vertices have different colors. There are many types of vertex coloring, such as complete coloring, k-differential coloring, and equitable coloring. Equitable coloring of G is a vertex coloring of G that satisfies the condition that for each induced color class it has an equitable cardinality with difference 0 or 1. The minimum number of colors used for such coloring of G is called the equitable chromatic number of G, denoted by χe(G). In this study, we only concern with graphs that have a central vertex, which means a vertex that is adjacent to every other vertex, in particular on the star graph (Sn), lollipop graph (Ln), and friendship graph (fn). This research aims to formulate the equitable chromatic number of the star graph (Sn), lollipop graph (Ln), and friendship graph (fn). The first step taken in this research is to apply vertex coloring to Sn, Ln, and fn. After that, the color classes of the vertex set are obtained and its cardinality is determined. Next, analyze that the applied vertex coloring meets the definition of equitable coloring. Then, prove that the number of colors used is minimum. Thus, the chromatic number for each graph is obtained and proved. Based on this research, the equitable chromatic number of Sn is ⌈n/2⌉ + 1, the equitable chromatic number of Ln is n, and the equitable chromatic number of fn is 3, for n = 1 and n + 1, for n ≥ 2.

An acyclic coloring of a graph G is a proper vertex coloring ( no two adjacent vertices of G have the same color) such that the induced sub graph of any two color classes is acyclic. The minimum number of colors required for acyclic coloring of a graph G is called as its acyclic chromatic number and is denoted by a(G). In this paper, we present the structure and coloring algorithm of central graph of Gear Graph Gn and we obtain the exact value of its acyclic chromatic number.

An edge of a properly vertex-colored graph is said to be a good edge if it has end vertices of different color. The chromatic completion graph of a graph G is a graph obtained by adding all possible good edges to G. The chromatic completion number of G is the maximum number of new good edges added to G. An equitable coloring of a graph G is a proper vertex coloring of G such that the difference of cardinalities of any two color classes is at most 1. In this paper, we discuss the chromatic completion graphs and chromatic completion number of certain graph classes, with respect to their equitable coloring.

Equitable colorings of planar graphs with maximum degree at least nine

An L-list coloring of a graph G is a proper vertex coloring in which every vertex v gets a color from a list L(v) of allowed colors. G is called k-choosable if all lists L(v) have exactly k elements and if G is L-list colorable for all possible assignments of such lists. Verifying conjectures of Erdos, Rubin and Taylor it was shown during the last years that every planar graph is 5-choosable and that there are planar graphs which are not 4-choosable. The question whether there are 3-colorable planar graphs which are not 4-choosable remained unsolved. The smallest known example far a non-4-choosable planar graph has 75 vertices and is described by Gutner. In fact, this graph is also 3 colorable and answers the above question. In addition, we give a list assignment for this graph using 5 colors only in all of the lists together such that the graph is not List-colorable. © 1997 John Wiley & Sons, Inc.

Indicated coloring of a graph G is a coloring in which there are two players Ann and Ben, Ann picks a vertex and Ben chooses a color for this vertex. The aim of Ann is to achieve a proper coloring of the whole graph G, while Ben tries to block the same. The smallest number of colors required for Ann to win the game on a graph G is called the indicated chromatic number of G and is denoted by \(\chi _i(G).\) In this paper, we prove that \(T\Box C_n, T\Box K_{n_1,n_2,\dots ,n_m}\) and \(K_{n_1,n_2,\dots ,n_m}\Box C_m\) are k-indicated colorable for all k greater than or equal to the indicated chromatic number of their corresponding Cartesian product, where T is any tree. Also we prove that \(\chi _i(K_{k_1,k_2,\dots ,k_m}\Box K_{l_1,l_2,\dots ,l_n}) =\chi (K_{k_1,k_2,\dots ,k_m}\Box K_{l_1,l_2,\dots ,l_n}).\) Finally we have given non-trivial examples of graphs G and H for which \(\chi _i(G\Box H) >\chi (G\Box H).\)

Often in problems with a parameter k like k-CNFSat and k-colorability, larger values of k make the problem harder. This is not always the case. Consider the problem of determining whether a planar graph has a k-coloring. The problem is trivial for k = 1, easy for k = 2 (check by DFS or BFS whether the graph is bipartite, i.e. has no odd cycles), and trivial for k = 4 or greater by the Four Color Theorem, which says that every planar graph is 4-colorable. This leaves k = 3. We show below that 3-colorability of planar graphs is no easier than 3-colorability of arbitrary graphs. This result is due to Garey, Johnson, and Stockmeyer [40]; see also Lichtenstein [72] for some other NP-completeness results involving planar graphs.