Injective Chromatic Number Research Articles

Injective coloring of subclasses of chordal graphs

An injective k-coloring of a graph G=(V,E) is a function f:V→{1,2,…,k} such that for every pair of vertices u and v having a common neighbor, f(u)≠f(v). The injective chromatic number χi(G) of a graph G is the minimum k for which G admits an injective k-coloring. Given a graph G and a positive integer k, Decide Injective Coloring Problem is to decide whether G admits an injective k-coloring. Decide Injective Coloring Problem is known to be NP-complete for chordal graphs. In this paper, we strengthen this result by proving that Decide Injective coloring Problem remains NP-complete for undirected path graphs, a proper subclass of chordal graphs. Moreover, we show that it is not possible to approximate the injective chromatic number of an undirected path graph within a factor of n1/3−ε in polynomial time for every ε>0 unless ZPP = NP. On the positive side, we prove that the injective chromatic number of an interval graph is either Δ(G) or Δ(G)+1, where Δ(G) is the maximum degree of G. We also characterize the interval graphs having χi(G)=Δ(G) and χi(G)=Δ(G)+1. As a consequence of this characterization, we obtain a linear time algorithm to find the injective chromatic number of an interval graph.

Injective coloring of graphs revisited

An open packing in a graph G is a set S of vertices in G such that no two vertices in S have a common neighbor in G. The injective chromatic number χi(G) of G is the smallest number of colors assigned to vertices of G such that each color class is an open packing. Alternatively, the injective chromatic number of G is the chromatic number of the two-step graph of G, which is the graph with the same vertex set as G in which two vertices are adjacent if they have a common neighbor. The concept of injective coloring has been studied by many authors, while in the present paper we approach it from two novel perspectives, related to open packings and the two-step graph operation. We prove several general bounds on the injective chromatic number expressed in terms of the open packing number. In particular, we prove that χi(G)≥12+14+2m−nρo holds for any connected graph G of order n≥2, size m, and the open packing number ρo, and characterize the class of graphs attaining the bound. Regarding the well known bound χi(G)≥Δ(G), we describe the family of extremal graphs and prove that deciding when the equality holds (even for regular graphs) is NP-complete, solving an open problem from an earlier paper. Next, we consider the chromatic number of the two-step graph of a graph, and compare it with the clique number and the maximum degree of the graph. We present two large families of graphs in which χi(G) equals the cardinality of a largest clique of the two-step graph of G. Finally, we consider classes of graphs that admit an injective coloring in which all color classes are maximal open packings. We give characterizations of three subclasses of these graphs among graphs with diameter 2, and find a partial characterization of hypercubes with this property.

List injective coloring of planar graphs

An injective coloring is a vertex coloring (not necessarily proper) such that any two vertices sharing a common neighbor receive distinct colors. A graph G is called injectively k-choosable, if for any color list L with admissible colors on V(G) of size k, there is an injective coloring φ such that φ(v)∈L(v) whenever v∈V(G). The list injective chromatic number, denoted by χil(G), is the least k for which G is injectively k-choosable. We focus on the study of list injective coloring on planar graphs which has disjoint 5−-cycles and show that χil(G)≤Δ+3 if Δ≥18 and χil(G)≤Δ+4 if Δ≥12.

List injective coloring of planar graphs with disjoint 5−-cycles

An injective [Formula: see text]-coloring of a graph [Formula: see text] is called injective if any two vertices joined by a path of length two get different colors. A graph [Formula: see text] is injectively [Formula: see text]-choosable if for any color list [Formula: see text] of admissible colors on [Formula: see text] of size [Formula: see text] it allows an injective coloring [Formula: see text] such that [Formula: see text] whenever [Formula: see text]. Let [Formula: see text], [Formula: see text] denote the injective chromatic number and injective choosability number of [Formula: see text], respectively. Let [Formula: see text] be a plane with disjoint [Formula: see text]-cycles and maximum degree [Formula: see text]. We show that [Formula: see text] if [Formula: see text], then [Formula: see text]; [Formula: see text] if [Formula: see text], then [Formula: see text].

Injective coloring of planar graphs

An injective k-coloring of a graph G is a mapping f:V(G)→{1,2,…,k} such that for any two vertices v1,v2∈V(G), f(v1)≠f(v2) if N(v1)∩N(v2)≠∅. The injective chromatic number of a graph G, denoted by χi(G), is the smallest integer k such that G has an injective k-coloring. In this paper, we prove that for a Halin graph G, χi(G)≤Δ(G)+2. Moreover, χi(G)≤Δ(G)+1 if Δ(G)≥6. Also, we show that for a triangle-free planar graph G without intersecting 4-cycles, χi(G)≤Δ(G)+6 if Δ(G)≥20.

Injective coloring of some subclasses of bipartite graphs and chordal graphs

A vertex coloring of a graph G=(V,E) that uses k colors is called an injectivek-coloring of G if no two vertices having a common neighbor have the same color. The minimum k for which G has an injectivek-coloring is called the injective chromatic number of G. Given a graph G and a positive integer k, the Decide Injective Coloring Problem is to decide whether G admits an injective k-coloring. It is known that Decide Injective Coloring Problem is NP-complete for bipartite graphs. In this paper, we strengthen this result by showing that this problem remains NP-complete for perfect elimination bipartite graphs, star-convex bipartite graphs and comb-convex bipartite graphs, which are proper subclasses of bipartite graphs. Moreover, we show that for every ϵ>0, it is not possible to efficiently approximate the injective chromatic number of a perfect elimination bipartite graph within a factor of n13−ϵ unless ZPP = NP. On the positive side, we propose a linear time algorithm for biconvex bipartite graphs and O(nm) time algorithm for convex bipartite graphs for finding the optimal injective coloring. We prove that the injective chromatic number of a chordal bipartite graph can be determined in polynomial time. It is known that Decide Injective Coloring Problem is NP-complete for chordal graphs. We give a linear time algorithm for computing the injective chromatic number of proper interval graphs, which is a proper subclass of chordal graphs. Decide Injective Coloring Problem is also known to be NP-complete for split graphs. We show that Decide Injective Coloring Problem remains NP-complete for K1,t-free split graphs for t≥4 and polynomially solvable for t≤3.

An Injective Version of the 1-2-3 Conjecture

In this work, we introduce and study a new graph labelling problem standing as a combination of the 1-2-3 Conjecture and injective colouring of graphs, which also finds connections with the notion of graph irregularity. In this problem, the goal, given a graph G, is to label the edges of G so that every two vertices having a common neighbour get incident to different sums of labels. We are interested in the minimum k such that G admits such a k-labelling. We suspect that almost all graphs G can be labelled this way using labels $$1,\dots ,\Delta (G)$$ . Towards this speculation, we provide bounds on the maximum label value needed in general. In particular, we prove that using labels $$1,\dots ,\Delta (G)$$ is indeed sufficient when G is a tree, a particular cactus, or when its injective chromatic number $$\mathrm{\chi _{\mathrm{i}}}(G)$$ is equal to $$\Delta (G)$$ .

Injective coloring of complementary prism and generalized complementary prism graphs

The complementary prism [Formula: see text] of a graph [Formula: see text] is the graph obtained by drawing edges between the corresponding vertices of a graph [Formula: see text] and its complement [Formula: see text]. In this paper, we generalize the concept of complementary prisms of graphs and determine the injective chromatic number of generalized complementary prisms of graphs. We prove that for any simple graph [Formula: see text] of order [Formula: see text], [Formula: see text] and if [Formula: see text] is a graph with a universal vertex, then [Formula: see text].

Injective chromatic number of lexicographic product of two graphs

Injective chromatic number of lexicographic product of two graphs

Injective Chromatic Number of Outerplanar Graphs

An injective coloring of a graph is a vertex coloring where two vertices with common neighbor receive distinct colors. The minimum integer $k$ such that $G$ has a $k$-injective coloring is called injective chromatic number of $G$ and denoted by $\chi_i(G)$. In this paper, the injective chromatic number of outerplanar graphs with maximum degree $\Delta$ and girth $g$ is studied. It is shown that every outerplanar graph $G$ has $\chi_i(G) \leq \Delta+2$, and this bound is tight. Then, it is proved that for an outerplanar graph $G$ with $\Delta = 3$, $\chi_i(G) \leq \Delta+1$ and the bound is tight for outerplanar graphs of girth $3$ and $4$. Finally, it is proved that, the injective chromatic number of $2$-connected outerplanar graphs with $\Delta = 3$, $g \geq 6$ and $\Delta \geq 4$, $g \geq 4$ is equal to $\Delta$.

Critical injective chromatic graphs

The concept of critical injective chromatic graph is introduced. Some properties of critical injective chromatic graph are studied based on number of vertices and degree of graphs. The condition for cycles to be critical injective chromatic is obtained. The existence of a critical injective chromatic graph is established. An algorithm for injective colouring of vertices of a given connected graph is proposed. An application of injective chromatic number and injective chromatic sum in the diagnosis of diseases based on symptoms is suggested.

On injective chromatic polynomials of graphs

The injective chromatic number χi(G) [G. Hahn, J. Kratochvil, J. Siran and D. Sotteau, On the injective chromatic number of graphs, Discrete Math. 256(1–2) (2002) 179–192] of a graph G is the minimum number of colors needed to color the vertices of G such that two vertices with a common neighbor are assigned distinct colors. The nature of the coefficients of injective chromatic polynomials of complete graphs, wheel graphs and cycles is studied. Injective chromatic polynomial on operations like union, join, product and corona of graphs is obtained.

Injective coloring of some graph operations

An injective coloring of a graph G is a vertex coloring such that any two vertices with a common vertex receive distinct colors. The injective chromatic number χi(G) of a graph G is the least k such that there is an injective k-coloring. Graph operations are important methods for constructing new graphs, and they play key roles in the design and analysis of networks. In this study, we give some sharp bounds (or exact values) of graph operations, including the Cartesian product, direct product, lexicographic product, union, join, and disjunction of graphs.

Injective Colorings with Arithmetic Constraints

An injective coloring of a graph is a vertex labeling such that two vertices sharing a common neighbor get different labels. In this work we introduce and study what we call additive colorings. An injective coloring $$c:V(G)\rightarrow \mathbb {Z}$$c:V(G)?Z of a graph $$G$$G is an additive coloring if for every $$uv, vw$$uv,vw in $$E(G)$$E(G), $$c(u)+c(w)\ne 2c(v)$$c(u)+c(w)?2c(v). The smallest integer $$k$$k such that an injective (resp. additive) coloring of a given graph $$G$$G exists with $$k$$k colors (resp. colors in $$\{1,\ldots ,k\}$${1,?,k}) is called the injective (resp. additive) chromatic number (resp. index). They are denoted by $$\chi _i(G)$$?i(G) and $$\chi '_a(G)$$?a?(G), respectively. In the first part of this work, we present several upper bounds for the additive chromatic index. On the one hand, we prove a super linear upper bound in terms of the injective chromatic number for arbitrary graphs, as well as a linear upper bound for bipartite graphs and trees. Complete graphs are extremal graphs for the super linear bound, while complete balanced bipartite graphs are extremal graphs for the linear bound. On the other hand, we prove a quadratic upper bound in terms of the maximum degree. In the second part, we study the computational complexity of computing $$\chi '_a(G)$$?a?(G). We prove that it can be computed in polynomial time for trees. We also prove that for bounded treewidth graphs, to decide whether $$\chi '_a(G)\le k$$?a?(G)≤k, for a fixed $$k$$k, can be done in polynomial time. On the other hand, we show that for cubic graphs it is NP-complete to decide whether $$\chi '_a(G)\le 4$$?a?(G)≤4. We also prove that for every $$\epsilon >0$$∈>0 there is a polynomial time approximation algorithm with approximation factor $$n^{1/3+\epsilon }$$n1/3+∈ for $$\chi '_a(G)$$?a?(G), when restricted to split graphs. However, unless $$\mathsf P =\mathsf NP $$P=NP, for every $$\epsilon >0$$∈>0 there is no polynomial time approximation algorithm with approximation factor $$n^{1/3-\epsilon }$$n1/3-∈ for $$\chi '_a(G)$$?a?(G), even when restricted to split graphs.

LIST INJECTIVE COLORING OF PLANAR GRAPHS WITH GIRTH g ≥ 5

An injective-k coloring of a graph G is a mapping cV(G) → {1, 2, …, k}, such that c(u) ≠ c(v) for each u, v ∈ V(G), whenever u, v have a common neighbor in G. If G has an injective-k coloring, then we call that G is injective-k colorable. Call χi(G) = min {k | G is injective-k colorable} is the injective chromatic number of G. Assign each vertex v ∈ V(G) a coloring set L(v), then L = {L(v) | v ∈ V(G)} is said to be a color list of G. Let L be a color list of G, if G has an injective coloring c such that c(v) ∈ L(v), ∀v ∈ V(G), then we call c an injective L-coloring of G. If for any color list L, such that |L(v)| ≥ k, G has an injective L-coloring, then G is said to be injective k-choosable. Call [Formula: see text] is injective k-choosable} is the injective chromatic number of G. So far, for the plane graph G of girth g(G) ≥ 5 and maximum degree Δ(G) ≥ 8, the best result of injective chromatic number is χi(G) ≤ Δ + 8. In this paper, for the plane graph G, we proved that [Formula: see text] if girth g(G) ≥ 5 and maximum degree Δ(G) ≥ 8.

Injective coloring of plane graphs with girth 5

An injective coloring of a graph is a vertex coloring where two vertices have distinct colors if a path of length two exists between them. Let G be a plane graph with g(G)≥5 and χi(G) be the injective chromatic number of G. In this paper, we improve some known results by proving that χi(G)≤Δ+3 if Δ(G)≥35 and χi(G)≤Δ+6 for arbitrary Δ(G).

Nordhaus–Gaddum-type relations of three graph coloring parameters

Let G be a simple graph. A coloring of vertices of G is called (i) a 2-proper coloring if vertices at distance 2 receive distinct colors; (ii) an injective coloring if vertices possessing a common neighbor receive distinct colors; (iii) a square coloring if vertices at distance at most 2 receive distinct colors. In this paper, we study inequalities of Nordhaus–Gaddum type for the 2-proper chromatic number, the injective chromatic number, and the square chromatic number.

Sufficient sparseness conditions for [formula omitted] to be [formula omitted]-choosable, when [formula omitted

We determine the list chromatic number of the square of a graph χℓ(G2) in terms of its maximum degree Δ when its maximum average degree, denoted mad(G), is sufficiently small. For Δ≥6, if mad(G)<2+4Δ−85Δ+2, then χℓ(G2)=Δ+1. In particular, if G is planar with girth g≥7+12Δ−2, then χℓ(G2)=Δ+1. Under the same conditions, χℓi(G)=Δ, where χℓi is the list injective chromatic number.

On the complexity of injective colorings and its generalizations

In this paper, we consider the complexity and algorithms of the injective coloring problem. We prove that it is NP-complete to determine the injective chromatic number even restricted to some special bipartite graphs. Furthermore, we show that for every ϵ>0, it is impossible to efficiently approximate the injective chromatic number of any bipartite graph within a factor of n13−ϵ unless ZPP=NP. For the max-injective coloring problem, we prove that there is a constant approximation algorithm on power chordal graphs with bounded injective chromatic number. We also devise a constant approximation algorithm for max-injective coloring some bipartite graphs. For the on-line injective coloring problem, we prove that First Fit (FF) injectively colors P3-free graphs perfectly, where First Fit is an on-line algorithm that simply assigns the smallest available color to each vertex. We also prove that the number of colors used by FF∗ for bipartite graph G is bounded by 32 times the on-line injective chromatic number, where FF∗ is an on-line algorithm equivalent to FF proper coloring the complement of G. Moreover, we present an improved algorithm BFF, and prove that it is optimal for on-line injectively coloring bipartite graphs.

Injective coloring of planar graphs with girth 6

An injective coloring of a graph is a vertex coloring where two vertices have distinct colors if a path of length two exists between them. Let χi(G) be the injective chromatic number of a graph G. In this paper, we investigate the injective coloring of planar graphs with girth 6. We improve some results of Borodin and Ivanova (2011) [1], Doyon et al. (2010) [4] and Li and Xu (2012) [6] by showing that if G is a planar graph with girth at least 6, then (1) χi(G)≤Δ+3; (2) χi(G)≤Δ+2 if Δ≥9; (3) χi(G)≤Δ+1 if Δ≥17.