Proper Vertex Coloring Research Articles

A counterexample to a conjecture on facial unique-maximal colorings

A facial unique-maximum coloring of a plane graph is a proper vertex coloring by natural numbers where on each face α the maximal color appears exactly once on the vertices of α. Fabrici and Göring (2016) proved that six colors are enough for any plane graph and conjectured that four colors suffice. This conjecture is a strengthening of the Four Color Theorem. Wendland (2016) later decreased the upper bound from six to five. In this note, we disprove the conjecture by giving an infinite family of counterexamples. Thus we conclude that the facial unique-maximum chromatic number of the sphere is five.

Dynamic coloring parameters for graphs with given genus

A proper vertex coloring of a graph G is r-dynamic if for each v∈V(G), at least min{r,d(v)} colors appear in NG(v). In this paper we investigate r-dynamic versions of coloring, list coloring, and paintability. We prove that planar and toroidal graphs are 3-dynamically 10-colorable, and this bound is sharp for toroidal graphs. We also give bounds on the minimum number of colors needed for any r in terms of the genus of the graph: for sufficiently large r, every graph with genus g is r-dynamically ((r+1)(g+5)+3)-colorable when g≤2 and r-dynamically ((r+1)(2g+2)+3)-colorable when g≥3. Furthermore, each of these upper bounds for r-dynamic k-colorability also holds for r-dynamic k-choosability and for r-dynamic k-paintability.

On Achromatic Coloring of Corona Graphs

Let \(G = (V(G),E(G))\) be a simple graph and an achromatic coloring of \(G\) is a proper vertex coloring of \(G\) in which every pair of colors appears on at least one pair of adjacent vertices. The achromatic number of \(G\) denoted by \(\psi(G)\), is the greatest number of colors in an achromatic coloring of \(G\). In this paper, we find out the achromatic number for Corona graph of Cycle with Path graphs on the same order \(n\), Path with Cycle graphs on the same order \(n\), Path with Complete graphs on the same order \(n\), Path of order \(n\) with Star graph on order \(n+1\), Path with Wheel graphs on the same order \(n\) and Ladder graph with Path graph on the same order \(n\).

On some domination colorings of graphs

In this paper, we are interested in four proper vertex colorings of graphs, with additional domination property. In the dominator colorings, strong colorings and strict strong colorings of a graph G, every vertex has to dominate at least one color class. Conversely, in the dominated colorings of G, every color class has to be dominated by at least one vertex. We study arbitrary graphs as well as P4-sparse graphs, P5-free graphs, bounded treewidth graphs and claw-free graphs.

Complexity of the Improper Twin Edge Coloring of Graphs

Let G be a graph whose each component has order at least 3. Let $$s : E(G) \rightarrow {\mathbb {Z}}_k$$ for some integer $$k\ge 2$$ be an improper edge coloring of G (where adjacent edges may be assigned the same color). If the induced vertex coloring $$c : V (G) \rightarrow {\mathbb {Z}}_k$$ defined by $$c(v) = \sum _{e\in E_v} s(e) \text{ in } {\mathbb {Z}}_k,$$ (where the indicated sum is computed in $${\mathbb {Z}}_k$$ and $$E_v$$ denotes the set of all edges incident to v) results in a proper vertex coloring of G, then we refer to such a coloring as an improper twin k-edge coloring. The minimum k for which G has an improper twin k-edge coloring is called the improper twin chromatic index of G and is denoted by $$\chi '_{it}(G)$$ . It is known that $$\chi '_{it}(G)=\chi (G)$$ , unless $$\chi (G) \equiv 2 \pmod 4$$ and in this case $$\chi '_{it}(G)\in \{\chi (G), \chi (G)+1\}$$ . In this paper, we first give a short proof of this result and we show that if G admits an improper twin k-edge coloring for some positive integer k, then G admits an improper twin t-edge coloring for all $$t\ge k$$ ; we call this the monotonicity property. In addition, we provide a linear time algorithm to construct an improper twin edge coloring using at most $$k+1$$ colors, whenever a k-vertex coloring is given. Then we investigate, to the best of our knowledge the first time in literature, the complexity of deciding whether $$\chi '_{it}(G)=\chi (G)$$ or $$\chi '_{it}(G)=\chi (G)+1$$ , and we show that, just like in case of the edge chromatic index, it is NP-hard even in some restricted cases. Lastly, we exhibit several classes of graphs for which the problem is polynomial.

Optimal Colorings with Rainbow Paths

Let G be a connected graph of chromatic number k. For a k-coloring f of G, a full f-rainbow path is a path of order k in G whose vertices are all colored differently by f. We show that G has a k-coloring f such that every vertex of G lies on a full f-rainbow path, which provides a positive answer to a question posed by Lin (Simple proofs of results on paths representing all colors in proper vertex-colorings, Graphs Combin. 23 2007, 201–203). Furthermore, we show that if G has a cycle of length 0 modulo k, then G has a k-coloring f such that, for every vertex u of G, some full f-rainbow path begins at u, which solves a problem posed by Bessy and Bousquet (Colorful paths for 3-chromatic graphs, arXiv:1503.00965v1 ) and verifies some special cases of a conjecture of Akbari et al. (Colorful paths in vertex coloring of graphs, preprint). Finally, we establish some more results on the existence of optimal colorings with (directed) full rainbow paths.

Color-blind index in graphs of very low degree

Let c:E(G)→[k] be an edge-coloring of a graph G, not necessarily proper. For each vertex v, let c̄(v)=(a1,…,ak), where ai is the number of edges incident to v with color i. Reorder c̄(v) for every v in G in nonincreasing order to obtain c∗(v), the color-blind partition of v. When c∗ induces a proper vertex coloring, that is, c∗(u)≠c∗(v) for every edge uv in G, we say that c is color-blind distinguishing. The minimum k for which there exists a color-blind distinguishing edge coloring c:E(G)→[k] is the color-blind index of G, denoted dal(G). We demonstrate that determining the color-blind index is more subtle than previously thought. In particular, determining if dal(G)≤2 is NP-complete. We also connect the color-blind index of a regular bipartite graph to 2-colorable regular hypergraphs and characterize when dal(G) is finite for a class of 3-regular graphs.

A flow based pruning scheme for enumerative equitable coloring algorithms

An equitable graph coloring is a proper vertex coloring of a graph G where the sizes of the color classes differ by at most one. The equitable chromatic number, denoted by $$\chi _{eq}(G),$$ is the smallest number k such that G admits such equitable k-coloring. We focus on enumerative algorithms for the computation of $$\chi _{eq}(G)$$ and propose a general scheme to derive pruning rules for them: We show how the extendability of a partial coloring into an equitable coloring can be modeled via network flows. Thus, we obtain pruning rules which can be checked via flow algorithms. Computational experiments show that the search tree of enumerative algorithms can be significantly reduced in size by these rules and, in most instances, such naive approach even yields a faster algorithm. Moreover, the stability, i.e., the number of solved instances within a given time limit, is greatly improved. Since the execution of flow algorithms at each node of a search tree is time consuming, we derive arithmetic pruning rules (generalized Hall-conditions) from the network model. Adding these rules to an enumerative algorithm yields an even larger runtime improvement.

Achromatic and Harmonious Colorings of Circulant Graphs

AbstractA proper vertex coloring of a graph G is achromatic (respectively harmonious) if every two colors appear together on at least one (resp. at most one) edge. The largest (resp. the smallest) number of colors in an achromatic (resp. a harmonious) coloring of G is called the achromatic (resp. harmonious chromatic) number of G and denoted by (resp. ). For a finite set of positive integers and a positive integer n, a circulant graph, denoted by , is an undirected graph on the set of vertices that has an edge if and only if either or is a member of (where substraction is computed modulo n). For any fixed set , we show that is asymptotically equal to , with the error term . We also prove that is asymptotically equal to , with the error term . As corollaries, we get results that improve, for a fixed k, the previously best estimations on the lengths of a shortest k‐radius sequence over an n‐ary alphabet (i.e., a sequence in which any two distinct elements of the alphabet occur within distance k of each other) and a longest packing k‐radius sequence over an n‐ary alphabet (which is a dual counterpart of a k‐radius sequence).

Degree choosable signed graphs

A signed graph is a graph in which each edge is labeled with +1 or −1. A (proper) vertex coloring of a signed graph is a mapping ϕ that assigns to each vertex v∈V(G) a color ϕ(v)∈Z such that every edge vw of G satisfies ϕ(v)≠σ(vw)ϕ(w), where σ(vw) is the sign of the edge vw. For an integer h≥0, let Z2h={±1,±2,…,±h} and Z2h+1=Z2h∪{0}. Following Máčajová et al. (2016), the chromatic number χ(G) of the signed graph G is the least integer k such that G admits a vertex coloring ϕ with im(ϕ)⊆Zk. As proved in Máčajová et al. (2016), every signed graph G satisfies χ(G)≤Δ(G)+1 and there are three types of signed connected simple graphs for which equality holds. We will extend this Brooks’ type result by considering graphs having multiple edges. We will also prove a list version of this result by characterizing degree choosable signed graphs. Furthermore, we will establish some basic facts about color critical signed graphs.

On chromatic Zagreb indices of certain graphs

Let [Formula: see text] be a finite and simple undirected connected graph of order [Formula: see text] and let [Formula: see text] be a proper vertex coloring of [Formula: see text]. Denote [Formula: see text] simply, [Formula: see text]. In this paper, we introduce a variation of the well-known Zagreb indices by utilizing the parameter [Formula: see text] instead of the invariant [Formula: see text] for all vertices of [Formula: see text]. The new indices are called chromatic Zagreb indices. We study these new indices for certain classes of graphs and introduce the notion of chromatically stable graphs.

Local Antimagic Vertex Coloring of a Graph

Let $$G=(V,E)$$G=(V,E) be a connected graph with $$\left| V \right| =n$$V=n and $$\left| E \right| = m.$$E=m. A bijection $$f:E \rightarrow \{1,2, \dots , m\}$$f:E?{1,2,?,m} is called a local antimagic labeling if for any two adjacent vertices u and v, $$w(u)\ne w(v),$$w(u)?w(v), where $$w(u)=\sum \nolimits _{e\in E(u)}{f(e)},$$w(u)=?e?E(u)f(e), and E(u) is the set of edges incident to u. Thus any local antimagic labeling induces a proper vertex coloring of G where the vertex v is assigned the color w(v). The local antimagic chromatic number $$\chi _{la}(G)$$?la(G) is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper we present several basic results on this new parameter.

Spotting Trees with Few Leaves

We show two results related to finding trees and paths in graphs. First, we show that in $O^*(1.657^k2^{l/2})$ time one can either find a $k$-vertex tree with $l$ leaves in an $n$-vertex undirected graph or conclude that such a tree does not exist. Our solution can be applied as a subroutine to solve the $k$-Internal Spanning Tree problem in $O^*(min(3.455^k, 1.946^n))$ time using polynomial space, improving upon previous algorithms for this problem. In particular, for the first time we break the natural barrier of $O^*(2^n)$. Second, we show that the running time can be improved whenever the host graph admits a vertex coloring with few colors; it can be an ordinary proper vertex coloring, a fractional vertex coloring, or a vector coloring. In effect, we show improved bounds for Hamiltonicity and $k$-Path in any graph of maximum degree $\Delta=4,\ldots,12$ or with vector chromatic number at most 8. Our results extend the technique by Bjorklund [SIAM J. Comput., 43 (2014), pp. 280--299] and Bjorklund et al...

Induced Colorful Trees and Paths in Large Chromatic Graphs

In a proper vertex coloring of a graph a subgraph is colorful if its vertices are colored with different colors. It is well-known (see for example in Gyárfás (1980)) that in every proper coloring of a $k$-chromatic graph there is a colorful path $P_k$ on $k$ vertices. The first author proved in 1987 that $k$-chromatic and triangle-free graphs have a path $P_k$ which is an induced subgraph. N.R. Aravind conjectured that these results can be put together: in every proper coloring of a $k$-chromatic triangle-free graph, there is an induced colorful $P_k$. Here we prove the following weaker result providing some evidence towards this conjecture: For a suitable function $f(k)$, in any proper coloring of an $f(k)$-chromatic graph of girth at least five, there is an induced colorful path on $k$ vertices.

Vertex-coloring [formula omitted]-edge-weighting of some graphs

Let G be a non-trivial graph and k∈Z+. A vertex-coloring k-edge-weighting is an assignment f:E(G)→{1,…,k} such that the induced labeling f:V(G)→Z+, where f(v)=∑e∈E(v)f(e) is a proper vertex coloring of G. It is proved in this paper that every4-edge-connected graph with chromatic number at most4admits a vertex-coloring3-edge-weighting.

The 3-flow conjecture, factors modulo k, and the 1-2-3-conjecture

Let k be an odd natural number ≥5, and let G be a (6k−7)-edge-connected graph of bipartite index at least k−1. Then, for each mapping f:V(G)→N, G has a subgraph H such that each vertex v has H-degree f(v) modulo k. We apply this to prove that, if c:V(G)→Zk is a proper vertex-coloring of a graph G of chromatic number k≥5 or k−1≥6, then each edge of G can be assigned a weight 1 or 2 such that each weighted vertex-degree of G is congruent to c modulo k. Consequently, each nonbipartite (6k−7)-edge-connected graph of chromatic number at most k (where k is any odd natural number ≥3) has an edge-weighting with weights 1,2 such that neighboring vertices have distinct weighted degrees (even after reducing these weighted degrees modulo k). We characterize completely the bipartite graph having an edge-weighting with weights 1,2 such that neighboring vertices have distinct weighted degrees. In particular, that problem belongs to P while it is NP-complete for nonbipartite graphs. The characterization also implies that every 3-edge-connected bipartite graph with at least 3 vertices has such an edge-labeling, and so does every simple bipartite graph of minimum degree at least 3.

Kempe equivalence of colourings of cubic graphs

Given a graph G=(V,E) and a proper vertex colouring of G, a Kempe chain is a subset of V that induces a maximal connected subgraph of G in which every vertex has one of two colours. To make a Kempe change is to obtain one colouring from another by exchanging the colours of vertices in a Kempe chain. Two colourings are Kempe equivalent if each can be obtained from the other by a series of Kempe changes. A conjecture of Mohar asserts that, for k≥3, all k-colourings of connected k-regular graphs that are not complete are Kempe equivalent. We address the case k=3 by showing that all 3-colourings of a connected cubic graph G are Kempe equivalent unless G is the complete graph K4 or the triangular prism.

The Chromatic Number of a Signed Graph

In 1982, Zaslavsky introduced the concept of a proper vertex colouring of a signed graph $G$ as a mapping $\phi\colon V(G)\to \mathbb{Z}$ such that for any two adjacent vertices $u$ and $v$ the colour $\phi(u)$ is different from the colour $\sigma(uv)\phi(v)$, where is $\sigma(uv)$ is the sign of the edge $uv$. The substantial part of Zaslavsky's research concentrated on polynomial invariants related to signed graph colourings rather than on the behaviour of colourings of individual signed graphs. We continue the study of signed graph colourings by proposing the definition of a chromatic number for signed graphs which provides a natural extension of the chromatic number of an unsigned graph. We establish the basic properties of this invariant, provide bounds in terms of the chromatic number of the underlying unsigned graph, investigate the chromatic number of signed planar graphs, and prove an extension of the celebrated Brooks' theorem to signed graphs.

Row and Column Extensions of 4-Cycle Free LDPC Codes

In this letter, an approach is proposed to increase the row (column)-weight of the parity-check matrix of a 4-cycle free LDPC code such that the constructed LDPC code has girth of at least 6. In fact, to each parity-check matrix $H$ , a new graph $G_r(H)$ ( $G_c(H)$ ) is assigned in which the vertices correspond to the rows (columns) of $H$ and two vertices are adjacent if and only if the associated rows (columns) have in common at least one nonzero element. Now, in a proper vertex coloring of $G_r(H)$ ( $G_c(H)$ ), each color is considered as a new column (row) whose nonzero elements happen in the rows (columns) in which the corresponding vertices have the same color. Based on this method, some high-rate LDPC codes with girth 6 and column-weight of at least 4 can be constructed from the recently proposed column-weight three LDPC codes with girth 6. Moreover, using the mutually orthogonal Latin squares, this approach is applied on the incidence matrices of some complete bipartite graphs to generate some girth-6 LDPC codes with different column-weights and large minimum-distances.

Ramsey numbers involving a long path

Let H be a graph, and σ(H) the minimum size of a color class among all proper vertex-coloring of H with χ(H) colors. A connected graph G with |G|≥σ(H) is said to be H-good if R(G,H)=(χ(H)−1)(|G|−1)+σ(H). In this note, we shall show that if n≥8|H|+3σ2(H)+cχ8(H), then Pn is H-good, where c=1014.