On-line and first fit colorings of graphs

A graph coloring algorithm that immediately colors the vertices taken from a list without looking ahead or changing colors already assigned is called “on-line coloring.” The properties of on-line colorings are investigated in several classes of graphs. In many cases we find on-line colorings that use no more colors than some function of the largest clique size of the graph. We show that the first fit on-line coloring has an absolute performance ratio of two for the complement of chordal graphs. We prove an upper bound for the performance ratio of the first fit coloring on interval graphs. It is also shown that there are simple families resisting any on-line algorithm: no on-line algorithm can color all trees by a bounded number of colors.     

Relaxed Locally Identifying Coloring of Graphs

A locally identifying coloring (lid-coloring) of a graph is a proper vertex-coloring such that the sets of colors appearing in the closed neighborhoods of any pair of adjacent vertices having distinct neighborhoods are distinct. Our goal is to study a relaxed locally identifying coloring (rlid-coloring) of a graph that is similar to locally identifying coloring for which the coloring is not necessarily proper. We denote by \(\chi _{rlid}(G)\) the minimum number of colors used in a relaxed locally identifying coloring of a graph G. In this paper, we prove that the problem of deciding that \(\chi _{rlid}(G)=3\) for a 2-degenerate planar graph G is NP-complete and for a bipartite graph G is polynomial. We give several bounds of \(\chi _{rlid}(G)\) for different families of graphs and construct new graphs for which these bounds are tight. We also compare this parameter with the minimum number of colors used in a locally identifying coloring of a graph G (\(\chi _{lid}(G)\)), the size of a minimum identifying code of G (\(\gamma _{id}(G)\)) and the chromatic number of G (\(\chi (G)\)).     

Achromatic number of fragmentable graphs

A complete coloring of a simple graph G is a proper vertex coloring such that each pair of colors appears together on at least one edge. The achromatic number ψ(G) is the greatest number of colors in such a coloring. We say a class of graphs is fragmentable if for any positive ε, there is a constant C such that any graph in the class can be broken into pieces of size at most C by removing a proportion at most ε of the vertices. Examples include planar graphs and grids of fixed dimension. Determining the achromatic number of a graph is NP‐complete in general, even for trees, and the achromatic number is known precisely for only very restricted classes of graphs. We extend these classes very considerably, by giving, for graphs in any class which is fragmentable, triangle‐free, and of bounded degree, a necessary and sufficient condition for a sufficiently large graph to have a complete coloring with a given number of colors. For the same classes, this gives a tight lower bound for the achromatic number of sufficiently large graphs, and shows that the achromatic number can be determined in polynomial time. As examples, we give exact values of the achromatic number for several graph families. © 2009 Wiley Periodicals, Inc. J Graph Theory 65:94–114, 2010     

Improved bounds on linear coloring of plane graphs

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest number of colors in a linear coloring of G. In this paper, we give some upper bounds on linear chromatic number for plane graphs with respect to their girth, that improve some results of Raspaud and Wang (2009).     

Acyclic edge coloring of subcubic graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and it is denoted by a^′(G). From a result of Burnstein it follows that all subcubic graphs are acyclically edge colorable using five colors. This result is tight since there are 3-regular graphs which require five colors. In this paper we prove that any non-regular connected graph of maximum degree 3 is acyclically edge colorable using at most four colors. This result is tight since all edge maximal non-regular connected graphs of maximum degree 3 require four colors.     

Upper bounds on vertex distinguishing chromatic index of some Halin graphs

A vertex distinguishing edge coloring of a graph G is a proper edge coloring of G such that any pair of vertices has the distinct sets of colors. The minimum number of colors required for a vertex distinguishing edge coloring of a graph G is denoted by ???? _s (G). In this paper, we obtained upper bounds on the vertex distinguishing chromatic index of 3-regular Halin graphs and Halin graphs with ??(G) ?? 4, respectively.     

Conditions on Ramsey Nonequivalence

Given a graph H , a graph G is called a Ramsey graph of H if there is a monochromatic copy of H in every coloring of the edges of G with two colors. Two graphs G , H are called Ramsey equivalent if they have the same set of Ramsey graphs. Fox et al. (J Combin Theory Ser B 109 (2014), 120–133) asked whether there are two nonisomorphic connected graphs that are Ramsey equivalent. They proved that a clique is not Ramsey equivalent to any other connected graph. Results of Ne?et?il et al. showed that any two graphs with different clique number (Combinatorica 1(2) (1981), 199–202) or different odd girth (Comment Math Univ Carolin 20(3) (1979), 565–582) are not Ramsey equivalent. These are the only structural graph parameters we know that “distinguish” two graphs in the above sense. This article provides further supportive evidence for a negative answer to the question of Fox et al. by claiming that for wide classes of graphs, the chromatic number is a distinguishing parameter. In addition, it is shown here that all stars and paths and all connected graphs on at most five vertices are not Ramsey equivalent to any other connected graph. Moreover, two connected graphs are not Ramsey equivalent if they belong to a special class of trees or to classes of graphs with clique‐reduction properties.     

Coloring, location and domination of corona graphs

A vertex coloring of a graph G is an assignment of colors to the vertices of G so that every two adjacent vertices of G have different colors. A coloring related property of a graphs is also an assignment of colors or labels to the vertices of a graph, in which the process of labeling is done according to an extra condition. A set S of vertices of a graph G is a dominating set in G if every vertex outside of S is adjacent to at least one vertex belonging to S. A domination parameter of G is related to those structures of a graph that satisfy some domination property together with other conditions on the vertices of G. In this article we study several mathematical properties related to coloring, domination and location of corona graphs. We investigate the distance-k colorings of corona graphs. Particularly, we obtain tight bounds for the distance-2 chromatic number and distance-3 chromatic number of corona graphs, through some relationships between the distance-k chromatic number of corona graphs and the distance-k chromatic number of its factors. Moreover, we give the exact value of the distance-k chromatic number of the corona of a path and an arbitrary graph. On the other hand, we obtain bounds for the Roman dominating number and the locating–domination number of corona graphs. We give closed formulaes for the k-domination number, the distance-k domination number, the independence domination number, the domatic number and the idomatic number of corona graphs.     

On resistance of graphs

An edge-coloring of a graph G with integers is called an interval coloring if all colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. It is known that not all graphs have interval colorings, and therefore it is expedient to consider a measure of closeness for a graph to be interval colorable. In this paper we introduce such a measure (resistance of a graph) and we determine the exact value of the resistance for some classes of graphs.     

若干图的倍图的均匀邻强边染色

如果图G的一个正常边染色满足相邻点的色集不同,且任意两种颜色所染边数目相差不超过1,则称为均匀邻强边染色,其所用最少染色数称为均匀邻强边色数．本文得到了星、扇和轮的倍图的均匀邻强边色数．     

Linear colorings of subcubic graphs

A linear coloring of a graph is a proper coloring of the vertices of the graph so that each pair of color classes induces a union of disjoint paths. In this paper, we prove that for every connected graph with maximum degree at most three and every assignment of lists of size four to the vertices of the graph, there exists a linear coloring such that the color of each vertex belongs to the list assigned to that vertex and the neighbors of every degree-two vertex receive different colors, unless the graph is _C5$C_5$ or _K3,3$K_{3,3}$. This confirms a conjecture raised by Esperet, Montassier and Raspaud [L. Esperet, M. Montassier, and A. Raspaud, Linear choosability of graphs, Discrete Math. 308 (2008) 3938–3950]. Our proof is constructive and yields a linear-time algorithm to find such a coloring.     

一些图的Mycielski图的均匀邻强边染色

如果图G的一个正常边染色满足相邻点的色集不同,且任意两种颜色所染边数目相差不超过1,则称为均匀邻强边染色,其所用最少染色数称为均匀邻强边色数.本文得到了路、圈、星和扇的Mycielski图的均匀邻强边色数.     

Acyclic colorings of graphs with bounded degree

A k-coloring (not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colors i and j the subgraph induced by the edges whose endpoints have colors i and j is acyclic. We consider some generalized acyclic k-colorings, namely, we require that each color class induces an acyclic or bounded degree graph. Mainly we focus on graphs with maximum degree 5. We prove that any such graph has an acyclic 5-coloring such that each color class induces an acyclic graph with maximum degree at most 4. We prove that the problem of deciding whether a graph G has an acyclic 2-coloring in which each color class induces a graph with maximum degree at most 3 is NP-complete, even for graphs with maximum degree 5. We also give a linear-time algorithm for an acyclic t-improper coloring of any graph with maximum degree d assuming that the number of colors is large enough.     

The harmonious coloring problem is NP-complete for interval and permutation graphs

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 previous work on the NP-completeness of the harmonious coloring problem when restricted to the class of disconnected graphs which are simultaneously cographs and interval graphs, we prove that the problem is also NP-complete for connected interval and permutation graphs.     

关于积图点可区别边染色的若干结论

如果图G的一个正常边染色满足任意两个不同点的关联边色集不同, 则称为点可区别边染色(VDEC), 其所用最少颜色数称为点可区别边色数. 利用构造法给出了积图点可区别边染色的一个结论, 得到了关于积图点可区别边色数的若干结果, 并且给出25个具体积图的点可区别边色数, 验证了它们满足点可区别边染色猜想(VDECC).     

Algorithms for finding distance-edge-colorings of graphs

For a bounded integer ℓ, we wish to color all edges of a graph G so that any two edges within distance ℓ have different colors. Such a coloring is called a distance-edge-coloring or an ℓ-edge-coloring of G. The distance-edge-coloring problem is to compute the minimum number of colors required for a distance-edge-coloring of a given graph G. A partial k-tree is a graph with tree-width bounded by a fixed constant k. We first present a polynomial-time exact algorithm to solve the problem for partial k-trees, and then give a polynomial-time 2-approximation algorithm for planar graphs.     

Mixed graph edge coloring

We are interested in coloring the edges of a mixed graph, i.e., a graph containing unoriented and oriented edges. This problem is related to a communication problem in job-shop scheduling systems. In this paper we give general bounds on the number of required colors and analyze the complexity status of this problem. In particular, we provide NP-completeness results for the case of outerplanar graphs, as well as for 3-regular bipartite graphs (even when only 3 colors are allowed, or when 5 colors are allowed and the graph is fully oriented). Special cases admitting polynomial-time solutions are also discussed.     

Bounds on vertex colorings with restrictions on the union of color classes

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 ? 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 ?. This generalizes some well‐known types of proper colorings like acyclic colorings (where j = 2 and ?is the collection of all even cycles) and star colorings (where j = 2 and ?consists 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 ?. 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(d^8/7) 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 (j_i, ?_i) (i = 1, …, l) that the union of any j_i color classes has no copy of any member of ?_i and obtain upper bounds on the corresponding chromatic numbers. © 2011 Wiley Periodicals, Inc. J Graph Theory 66: 213–234, 2011     

Compact cyclic edge-colorings of graphs

The paper is devoted to a model of compact cyclic edge-coloring of graphs. This variant of edge-coloring finds its applications in modeling schedules in production systems, in which production proceeds in a cyclic way. We point out optimal colorings for some graph classes and we construct graphs which cannot be colored in a compact cyclic manner. Moreover, we prove some theoretical properties of considered coloring model such as upper bounds on the number of colors in optimal compact cyclic coloring.     

Inequalities for the Grundy chromatic number of graphs

The Grundy (or First-Fit) chromatic number of a graph G is the maximum number of colors used by the First-Fit coloring of the graph G. In this paper we give upper bounds for the Grundy number of graphs in terms of vertex degrees, girth, clique partition number and for the line graphs. Next we show that if the Grundy number of a graph is large enough then the graph contains a subgraph of prescribed large girth and Grundy number.