K-coloring Of Graph Research Articles

On Fall-Colorable Graphs

A fall k-coloring of a graph G is a proper k-coloring of G such that each vertex has at least one neighbor in each of the other color classes. A graph G which has a fall k-coloring is equivalent to having a partition of the vertex set V(G) in k independent dominating sets. In this paper, we first prove that for any fall k-colorable graph G with order n, the number of edges of G is at least (n(k−1)+r(k−r))/2, where r≡n(modk) and 0≤r≤k−1, and the bound is tight. Then, we obtain that if G is k-colorable (k≥2) and the minimum degree of G is at least k−2k−1n, then G is fall k-colorable and this condition of minimum degree is the best possible. Moreover, we give a simple proof for an NP-hard result of determining whether a graph is fall k-colorable, where k≥3. Finally, we show that there exist an infinite family of fall k-colorable planar graphs for k∈{5,6}.

Large independent sets in triangle-free cubic graphs: beyond planarity

The _independence ratio_ of a graph is the ratio of the size of its largest independent set to its number of vertices. Trivially, the independence ratio of a k-colorable graph is at least $1/k$ as each color class of a k-coloring is an independent set. However, better bounds can often be obtained for well-structured classes of graphs. In particular, Albertson, Bollobás and Tucker conjectured in 1976 that the independence ratio of every triangle-free subcubic planar graph is at least $3/8$. The conjecture was proven by Heckman and Thomas in 2006, and the ratio is best possible as there exists a cubic triangle-free planar graph with 24 vertices and the independence number equal to 9. The present article removes the planarity assumption. However, one needs to introduce an additional assumption since there are known to exist six 2-connected (non-planar) triangle-free subcubic graphs with the independence ratio less than $3/8$. Bajnok and Brinkmann conjectured that every 2-connected triangle-free subcubic graph has the independence ratio at least $3/8$ unless it is one of the six exceptional graphs. Fraughnaugh and Locke proposed a stronger conjecture: every triangle-free subcubic graph that does not contain one of the six exceptional graphs as a subgraph has independence ratio at least $3/8$. The authors prove these two conjectures, which implies in particular the result by Heckman and Thomas.

Parameterized complexity of happy coloring problems

In a vertex-colored graph, an edge is happy if its endpoints have the same color. Similarly, a vertex is happy if all its incident edges are happy. Motivated by the computation of homophily in social networks, we consider the algorithmic aspects of the following Maximum Happy Edges ( k-MHE ) problem: given a partially k-colored graph G and an integer ℓ, find an extended full k-coloring of G making at least ℓ edges happy. When we want to make ℓ vertices happy on the same input, the problem is known as Maximum Happy Vertices ( k-MHV ). We perform an extensive study into the complexity of the problems, particularly from a parameterized viewpoint. For every k≥3, we prove both problems can be solved in time 2nnO(1). Moreover, by combining this result with a linear vertex kernel of size (k+ℓ) for k-MHE, we show that the edge-variant can be solved in time 2ℓnO(1). In contrast, we prove that the vertex-variant remains W[1]-hard for the natural parameter ℓ. However, the problem does admit a kernel with O(k2ℓ2) vertices for the combined parameter k+ℓ. From a structural perspective, we show both problems are fixed-parameter tractable for treewidth and neighborhood diversity, which can both be seen as sparsity and density measures of a graph. Finally, we extend the known ▪-completeness results of the problems by showing they remain hard on bipartite graphs and split graphs. On the positive side, we show the vertex-variant can be solved optimally in polynomial-time for cographs.

The number of Gallai k-colorings of complete graphs

An edge coloring of the n-vertex complete graph, Kn, is a Gallai coloring if it does not contain any rainbow triangle, that is, a triangle whose edges are colored with three distinct colors. We prove that for n large and every k with k≤2n/4300, the number of Gallai colorings of Kn that use at most k given colors is ((k2)+on(1))2(n2). Our result is asymptotically best possible and implies that, for those k, almost all Gallai k-colorings use only two colors. However, this is not true for k≥2n/2.

Generalized Graph k-Coloring Games

We investigate pure Nash equilibria in generalized graphk-coloring games where we are given an edge-weighted undirected graph together with a set of k colors. Nodes represent players and edges capture their mutual interests. The strategy set of each player consists of k colors. The utility of a player v in a given state or coloring is given by the sum of the weights of edges {v, u} incident to v such that the color chosen by v is different than the one chosen by u, plus the profit gained by using the chosen color. Such games form some of the basic payoff structures in game theory, model lots of real-world scenarios with selfish players and extend or are related to several fundamental class of games. We first show that generalized graph k-coloring games are potential games. In particular, they are convergent and thus Nash Equilibria always exist. We then evaluate their performance by means of the widely used notions of price of anarchy and price of stability and provide tight bounds for two natural and widely used social welfare, i.e., utilitarian and egalitarian social welfare.

Heuristic method to determine lucky k-polynomials for k-colorable graphs

Abstract The existence of edges is a huge challenge with regards to determining lucky k-polynomials of simple connected graphs in general. In this paper the lucky 3-polynomials of path and cycle graphs of order, 3 ≤ n ≤ 8 are presented as the basis for the heuristic method to determine the lucky k-polynomials for k-colorable graphs. The difficulty of adjacency with graphs is illustrated through these elementary graph structures. The results are also illustratively compared with the results for null graphs (edgeless graphs). The paper could serve as a basis for finding recurrence results through innovative methodology.

The r-dynamic chromatic number of corona of order two of any graphs with complete graph

The corona of two graphs G and H is a graph G ⊙ H formed from one copy of G and |V (G)| copies of H where the ith vertex of G is adjacent to every vertex in the ith copy of H. For any integer l ≤ 2, we define the graph G ⊙l H recursively from G ⊙ H as G ⊙l H = (G ⊙l–1 H) ⊙ H. Graph G ⊙l H is also named as l-corona product of G and H. The r-dynamic chromatic number is the minimum k such that graph G has an r-dynamic k-coloring, denoted by χr(G). An r-dynamic coloring of a graph G is a proper k-coloring of graph G such that the neighbors of any vertex v receive at least min{r, d(v)} different colors. In this paper, we analyze the r-dynamic chromatic number of corona order two graph, denoted by G ⊙2 H; G is complete graph and H are any graph.

Some completion problems for graphs without chordless cycles of prescribed lengths

Given a proper vertex coloring of a graph G, a supergraph that remains properly colored is called a completion of G. In this paper, we first consider the problem of determining whether a given properly colored graph has a C4-free completion, and completely classify the complexity of this problem based on the number of colors used on the input graph. We show that this problem is polynomial time solvable when three colors are used, but NP-complete when the graph is colored with k≥4 colors. We then show that an identical dichotomy exists for the problem of completing a colored graph to be (C4,C5)-free. In contrast, we also show that for any fixed s≥1 the problem of determining if a k-colored graph has a (C4,C5,…,Cs+5)-free completion and the problem of determining if a k-colored graph can be completed to be (C4,C6,…,C2s+4)-free are NP-complete when the input graph is k-colored for any fixed k, k≥3.

Maximal ambiguously k-colorable graphs

A graph is ambiguously k-colorable if its vertex set admits two distinct partitions each into at most k anticliques. We give a full characterization of the maximal ambiguously k-colorable graphs in terms of k×k-matrices. As an application, we calculate the maximum number of edges an ambiguously k-colorable graph can have, and characterize the extremal graphs.

On r-dynamic chromatic number of coronation of order two of any graphs with path graph

In this study we assume every graphs be a simple, finite, nontrivial, connected and undirected graphs with vertex set V(H), edge set E(H) and no isolated vertex. An r-dynamic coloring of a graph H is a proper k-coloring of graph H such that the neighbors of any vertex u accept at least min{r, d(u)} different colors. The r-dynamic chromatic number is the minimum k such that graph H has an r-dynamic k-coloring, it is shown by χr(H). In this research, we interpret the r-dynamic chromatic number of graph corona of order two, denoted by H1⊙2 H2. The corona of two graphs H1 and H2 is a graph H1 ⊙ H2 formed from one copy of H1 and |V(H1)| copies of H2 where the ith vertex of H1 is adjacent to every vertex in the ith copy of H2. For any integer l ≥ 2, we establish the graph H1 ⊙l H2 recursively from H1 ⊙ H2 as H1 ⊙l H2 = (H1 ⊙l−1 H2) ⊙ H2. Graph H1 ⊙l H2 is also named as l − corona product of H1 and H2. Xr = (Pn ⊙2 H2) and H2 is path graph Pm, star graph Sm, complete graph Km and fan graph Fm. In this study we will determine the lower bound of r-dynamic chromatic number of corona order two of graphs.

Local antimagic vertex dynamic coloring of some graphs family

All graphs in this paper are simple and connected graph. A vertex dynamic coloring is a proper vertex k-coloring of graph G such that |c(N(vi))| ≥ min{r, d(v)} and the neighbourhood of vertex u has different colors. A bijection f : E(G) → {1, 2, 3, …, m} is called a local antimagic dynamic coloring, such that: (1) if uv ϵ E(G), where w(u) = ∑eϵE(u) f(e) and (2) for each vertex v ϵ V(G), |w(N(vi))| ≥ min{r, d(vi)}. The local antimagic vertex dynamic chromatic number denoted by is the minimum number of colors needed to color G in such a way the graph G to be local antimagic vertex dynamic graph. In this paper, we will study the existence of the local antimagic vertex dynamic chromatic number of some graph classes, namely caterpilar, doublebroom, broom and sun graph.

Local antimagic r-dynamic coloring of graphs

Let G = (V, E) be a connected graph. A bijection function f : E(G) → {1,2, 3, · · ·, E(G)|} is called a local antimagic labeling if for all uv ∈ E(G)s, w(u) ≠ w(v), where w(u) = ∑e∈E(u)f(e). Such that, local antimagic labeling induces a proper vertex k-coloring of graph G that the neighbors of any vertex u receive at least min{r, d(v)} different colors. The local antimagic r-dynamic chromatic number, denoted by (G) is the minimum k such that graph G has the local antimagic r-dynamic vertex k-coloring. In this paper, we will present the basic results namely the upper bound of the local antimagic r-dynamic chromatic number of some classes graph.

On the r-dynamic chromatic number of the corronation by complete graph

In this paper we will study the r-dynamic chromatic number of the coronation by complete graph. A proper k-coloring of graph G such that the neighbors of any vertex v receive at least min{r, d(v)} different colors. The r-dynamic chromatic number, χr(G) is the minimum k such that graph G has an r-dynamic k-coloring. We will obtain lower bound of the r-dynamic chromatic number of , and We also study the exact value of the r-dynamic chromatic number of and for m, n > 3.

Asymptotic multipartite version of the Alon–Yuster theorem

In this paper, we prove the asymptotic multipartite version of the Alon–Yuster theorem, which is a generalization of the Hajnal–Szemerédi theorem: If k≥3 is an integer, H is a k-colorable graph and γ>0 is fixed, then, for every sufficiently large n, where |V(H)| divides n, and for every balanced k-partite graph G on kn vertices with each of its corresponding (k2) bipartite subgraphs having minimum degree at least (k−1)n/k+γn, G has a subgraph consisting of kn/|V(H)| vertex-disjoint copies of H.The proof uses the Regularity method together with linear programming.

Completing colored graphs to meet a target property

We consider the problem of deciding whether a k-colored graph can be completed to have a given property. We establish that, when k is not fixed, the completion problems for (Helly) circular-arc graphs, even (Helly) proper or (Helly) unit circular-arc graphs, are NP-complete. When k is fixed, in the case of completion to a (Helly) circular-arc graph, (Helly) proper circular-arc graph, or (Helly) unit circular-arc graph we fully classify the complexities of the problems. We also show that deciding whether a 3-colored graph can be completed to be strongly chordal can be done in O(n2) time. As a corollary of our results, the sandwich problems for Helly circular-arc graphs, Helly proper circular-arc graphs, and Helly unit circular-arc graphs are NP-complete.

Size of edge-critical uniquely 3-colorable planar graphs

A graph G is uniquely k-colorable if the chromatic number of G is k and G has only one k-coloring up to permutation of the colors. A uniquely k-colorable graph G is edge-critical if G−e is not a uniquely k-colorable graph for any edge e∈E(G). Mel’nikov and Steinberg (Mel’nikov and Steinberg, 1977) asked to find an exact upper bound for the number of edges in an edge-critical uniquely 3-colorable planar graph with n vertices. In this paper, we give some properties of edge-critical uniquely 3-colorable planar graphs and prove that if G is such a graph with n(≥6) vertices, then |E(G)|≤52n−6, which improves the upper bound 83n−173 given by Matsumoto (Matsumoto, 2013). Furthermore, we find some edge-critical uniquely 3-colorable planar graphs with n(=10,12,14) vertices and 52n−7 edges.

Hardness and inapproximability of convex recoloring problems

Given a graph with an arbitrary vertex coloring, the Convex Recoloring Problem (CR) consists in recoloring the minimum number of vertices so that each color induces a connected subgraph. We focus on the complexity and inapproximability of this problem on k-colored graphs, for fixed k⩾2. We prove a strong complexity result showing that, for each k⩾2, CR is already NP-hard on k-colored grids, and therefore also on planar graphs with maximum degree 4. For each k⩾2, we prove that, for a positive constant c, there is no clnn-approximation algorithm for k-colored n-vertex (bipartite) graphs, unless P=NP. We also prove that CR parameterized by the number of color changes is W[2]-hard. For 2-colored (q,q−4)-graphs, a class that includes cographs and P4-sparse graphs, we present linear-time algorithms for fixed q. The same complexity and inapproximability results are obtained for two relaxations of the problem, where only one fixed color or any color is required to induce a connected subgraph, respectively.

Closing complexity gaps for coloring problems on H-free graphs

If a graph G contains no subgraph isomorphic to some graph H, then G is called H-free. A coloring of a graph G=(V,E) is a mapping c:V→{1,2,…} such that no two adjacent vertices have the same color, i.e., c(u)≠c(v) if uv∈E; if |c(V)|⩽k then c is a k-coloring. The Coloring problem is to test whether a graph has a coloring with at most k colors for some integer k. The Precoloring Extension problem is to decide whether a partial k-coloring of a graph can be extended to a k-coloring of the whole graph for some integer k. The List Coloring problem is to decide whether a graph allows a coloring, such that every vertex u receives a color from some given set L(u). By imposing an upper bound ℓ on the size of each L(u) we obtain the ℓ-List Coloring problem. We first classify the Precoloring Extension problem and the ℓ-List Coloring problem for H-free graphs. We then show that 3-List Coloring is NP-complete for n-vertex graphs of minimum degree n−2, i.e., for complete graphs minus a matching, whereas List Coloring is fixed-parameter tractable for this graph class when parameterized by the number of vertices of degree n−2. Finally, for a fixed integer k>0, the Listk-Coloring problem is to decide whether a graph allows a coloring, such that every vertex u receives a color from some given set L(u) that must be a subset of {1,…,k}. We show that List 4-Coloring is NP-complete for P6-free graphs, where P6 is the path on six vertices. This completes the classification of Listk-Coloring for P6-free graphs.

Constructing Uniquely Realizable Graphs

In the Graph Realization Problem (GRP), one is given a graph $$G$$ , a set of non-negative edge-weights, and an integer $$d$$ . The goal is to find, if possible, a realization of $$G$$ in the Euclidian space $$\mathbb R ^d$$ , such that the distance between any two vertices is the assigned edge weight. The problem has many applications in mathematics and computer science, but is NP-hard when the dimension $$d$$ is fixed. Characterizing tractable instances of GRP is a classical problem, first studied by Menger in 1931. We construct two new infinite families of GRP instances which can be solved in polynomial time. Both constructions are based on the blow-up of fixed small graphs with large expanders. Our main tool is the Connelly’s condition in Rigidity Theory, combined with an explicit construction and algebraic calculations of the rigidity (stress) matrix. As an application of our results, we describe a general framework to construct uniquely k-colorable graphs. These graphs have the extra property of being uniquely vector k-colorable. We give a deterministic explicit construction of such a family using Cayley expander graphs.

BILANGAN KROMATIK LOKASI DARI GRAF P m P n ; K m P n ; DAN K , m K n

Let G and H be two connected graphs. Let c be a vertex k-coloring of aconnected graph G and let = fCg be a partition of V (G) into the resultingcolor classes. For each v 2 V (G), the color code of v is dened to be k-vector: c1; C2; :::; Ck(v) =(d(v; C1); d(v; C2); :::; d(v; Ck)), where d(v; Ci) = minfd(v; x) j x 2 Cg, 1 i k. Ifdistinct vertices have distinct color codes with respect to , then c is called a locatingcoloring of G. The locating chromatic number of G is the smallest natural number ksuch that there are locating coloring with k colors in G. The Cartesian product of graphG and H is a graph with vertex set V (G) V (H), where two vertices (a; b) and (a)are adjacent whenever a = a0and bb02 E(H), or aa0i2 E(G) and b = b, denotedby GH. In this paper, we will study about the locating chromatic numbers of thecartesian product of two paths, the cartesian product of paths and complete graphs, andthe cartesian product of two complete graphs.