Minimum Number Of Colors Research Articles

Excluding a Ladder

A ladder is a $2 \times k$ grid graph. When does a graph class $\mathcal{C}$ exclude some ladder as a minor? We show that this is the case if and only if all graphs $G$ in $\mathcal{C}$ admit a proper vertex coloring with a bounded number of colors such that for every $2$-connected subgraph $H$ of $G$, there is a color that appears exactly once in $H$. This type of vertex coloring is a relaxation of the notion of centered coloring, where for every connected subgraph $H$ of $G$, there must be a color that appears exactly once in $H$. The minimum number of colors in a centered coloring of $G$ is the treedepth of $G$, and it is known that classes of graphs with bounded treedepth are exactly those that exclude a fixed path as a subgraph, or equivalently, as a minor. In this sense, the structure of graphs excluding a fixed ladder as a minor resembles the structure of graphs without long paths. Another similarity is as follows: It is an easy observation that every connected graph with two vertex-disjoint paths of length $k$ has a path of length $k+1$. We show that every $3$-connected graph which contains as a minor a union of sufficiently many vertex-disjoint copies of a $2 \times k$ grid has a $2 \times (k+1)$ grid minor. Our structural results have applications to poset dimension. We show that posets whose cover graphs exclude a fixed ladder as a minor have bounded dimension. This is a new step towards the goal of understanding which graphs are unavoidable as minors in cover graphs of posets with large dimension.

Chromatic number of super vertex local antimagic total labelings of graphs

Let G(V,E) be a simple graph and f be a bijection f : V ∪ E → {1, 2, …, | V |+| E |} where f ( V )={1, 2, …, | V |} . For a vertex x ∈ V , define its weight w ( x ) as the sum of labels of all edges incident with x and the vertex label itself. Then f is called a super vertex local antimagic total (SLAT) labeling if for every two adjacent vertices their weights are different. The super vertex local antimagic total chromatic number χ s l a t ( G ) is the minimum number of colors taken over all colorings induced by super vertex local antimagic total labelings of G. We classify all trees T that have χ s l a t ( T )=2 , present a class of trees that have χ s l a t ( T )=3 , and show that for any positive integer n ≥ 2 there is a tree T with χ s l a t ( T )= n .

Conflict‐free chromatic number versus conflict‐free chromatic index

AbstractA vertex coloring of a given graph is conflict‐free if the closed neighborhood of every vertex contains a unique color (i.e., a color appearing only once in the neighborhood). The minimum number of colors in such a coloring is the conflict‐free chromatic number of , denoted . What is the maximum possible conflict‐free chromatic number of a graph with a given maximum degree ? Trivially, , but it is far from optimal—due to results of Glebov, Szabó, and Tardos, and of Bhyravarapu, Kalyanasundaram, and Mathew, the answer is known to be . We show that the answer to the same question in the class of line graphs is —it follows that the extremal value of the conflict‐free chromatic index among graphs with maximum degree is much smaller than the one for conflict‐free chromatic number. The same result for is also provided in the class of near regular graphs, that is, graphs with minimum degree .

Tight Bounds on the Clique Chromatic Number

The clique chromatic number of a graph is the minimum number of colours needed to colour its vertices so that no inclusion-wise maximal clique which is not an isolated vertex is monochromatic. We show that every graph of maximum degree $\Delta$ has clique chromatic number $O\left(\frac{\Delta}{\log~\Delta}\right)$. We obtain as a corollary that every $n$-vertex graph has clique chromatic number $O\left(\sqrt{\frac{n}{\log ~n}}\right)$. Both these results are tight.

On Asymmetric Colourings of Claw-Free Graphs

A vertex colouring of a graph is asymmetric if it is preserved only by the identity automorphism. The minimum number of colours needed for an asymmetric colouring of a graph $G$ is called the asymmetric colouring number or distinguishing number $D(G)$ of $G$. It is well known that $D(G)$ is closely related to the least number of vertices moved by any non-identity automorphism, the so-called motion $m(G)$ of $G$. Large motion is usually correlated with small $D(G)$. Recently, Babai posed the question whether there exists a function $f(d)$ such that every connected, countable graph $G$ with maximum degree $\Delta(G)\leq d$ and motion $m(G)>f(d)$ has an asymmetric $2$-colouring, with at most finitely many exceptions for every degree.
 We prove the following result: if $G$ is a connected, countable graph of maximum degree at most 4, without an induced claw $K_{1,3}$, then $D(G)= 2$ whenever $m(G)>2$, with three exceptional small graphs. This answers the question of Babai for $d=4$ in the class of~claw-free graphs.

List injective edge-coloring of subcubic graphs

An edge-coloring of a graph G is injective if for any two distinct edges e1 and e2, the colors of e1 and e2 are distinct if they are at distance 2 in G or in a common triangle. The injective chromatic index of G, χinj′(G), is the minimum number of colors needed for an injective edge-coloring of G. In this paper, we consider the list version of the injective edge-coloring and prove that the list-injective chromatic index of every subcubic graph is at most 7, which generalizes the recent result of Kostochka et al. (2021).

On the double total dominator chromatic number of graphs

In this paper, we introduce and study a new coloring problem of graphs called the double total dominator coloring. A double total dominator coloring of a graph [Formula: see text] with minimum degree at least 2 is a proper vertex coloring of [Formula: see text] such that each vertex has to dominate at least two color classes. The minimum number of colors among all double total dominator coloring of [Formula: see text] is called the double total dominator chromatic number, denoted by [Formula: see text]. Therefore, we establish the close relationship between the double total dominator chromatic number [Formula: see text] and the double total domination number [Formula: see text]. We prove the NP-completeness of the problem. We also examine the effects on [Formula: see text] when [Formula: see text] is modified by some operations. Finally, we discuss the [Formula: see text] number of square of trees by giving some bounds.

Proof of a conjecture on the strong chromatic index of Halin graphs

A strong edge coloring of a graph G is an assignment of colors to the edges of G such that two distinct edges are colored differently if their distance is at most two. The strong chromatic index of graph G, denoted by χs′(G), is the minimum number of colors needed for a strong edge coloring of G. A Halin graph G is a plane graph constructed from a tree T without vertices of degree two by connecting all leaves through a cycle C. In 2018, Hu et al. (2018) posed a conjecture which says that if G=T∪C is a Halin graph other than a wheel Wn, Ne2, or Ne4, then χs′(G)≤χs′(T)+2. The bound can be achieved. In this paper, we show that the above conjecture is true. This improves the known bound χs′(G)≤χs′(T)+3 obtained by Lai et al. (2012), and extends the results on Halin graphs G with Δ(G)≤3 obtained by Lih and Liu (2012) and Δ(G)≤4 obtained by Hu et al. (2018), respectively.

Color Energy of Some Cluster Graphs

Let $G$ be a simple connected graph. The energy of a graph $G$ is defined as sum of the absolute eigenvalues of an adjacency matrix of the graph $G$. It represents a proper generalization of a formula valid for the total $\pi$-electron energy of a conjugated hydrocarbon as calculated by the Huckel molecular orbital (HMO) method in quantum chemistry. A coloring of a graph $G$ is a coloring of its vertices such that no two adjacent vertices share the same color. The minimum number of colors needed for the coloring of a graph $G$ is called the chromatic number of $G$ and is denoted by $\chi(G)$. The color energy of a graph $G$ is defined as the sum of absolute values of the color eigenvalues of $G$. The graphs with large number of edges are referred as cluster graphs. Cluster graphs are graphs obtained from complete graphs by deleting few edges according to some criteria. It can be obtained on deleting some edges incident on a vertex, deletion of independent edges/triangles/cliques/path P3 etc. Bipartite cluster graphs are obtained by deleting few edges from complete bipartite graphs according to some rule. In this paper, the color energy of cluster graphs and bipartite cluster graphs are studied.

On Locating Chromatic Number of Cubic Graph with Tree Cycle, Cn,2n,n, for n=3,4,5

Let G = (V(G), E(G)) be a connected graph and is coloring of graph G. Let Π = {C 1, C 2, …,Ck }, where Ci is the partition of the vertex in which is colored i with 1 ≥ i ≥ k. The representation v for Π is called the color code, denoted C Π(v) is a ordered pair with k-element namely, C Π(v) = (d(v, C 1), d(v, C 2), …, d(v, Ck )), where d(v, Ci )= mind{d(v, x)|xεCi } for 1 ≥ i ≥ k. If every vertex in G have different color code, the c is locating coloring. The minimum number of colors used in G is called chromatic locating, notated by XL (G). In this paper, we will determine the locating coloring of graph cubic Cn,2n,n , for n=3,4,5.

Local Antimagic Chromatic Number for Copies of Graphs

An edge labeling of a graph G=(V,E) using every label from the set {1,2,⋯,|E(G)|} exactly once is a local antimagic labeling if the vertex-weights are distinct for every pair of neighboring vertices, where a vertex-weight is the sum of labels of all edges incident with that vertex. Any local antimagic labeling induces a proper vertex coloring of G where the color of a vertex is its vertex-weight. This naturally leads to the concept of a local antimagic chromatic number. The local antimagic chromatic number is defined to be the minimum number of colors taken over all colorings of G induced by local antimagic labelings of G. In this paper, we estimate the bounds of the local antimagic chromatic number for disjoint union of multiple copies of a graph.

Extension of Gyárfás-Sumner Conjecture to Digraphs

The dichromatic number of a digraph $D$ is the minimum number of colors needed to color its vertices in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has become the focus of numerous works. In this work we look at possible extensions of the Gyárfás-Sumner conjecture. In particular, we conjecture a simple characterization of sets $\mathcal F$ of three digraphs such that every digraph with sufficiently large dichromatic number must contain a member of $\mathcal F$ as an induced subdigraph. 
 Among notable results, we prove that oriented $K_4$-free graphs without a directed path of length $3$ have bounded dichromatic number where a bound of $414$ is provided. We also show that an orientation of a complete multipartite graph with no directed triangle is $2$-colorable. To prove these results we introduce the notion of nice sets that might be of independent interest.

Total colorings of certain classes of lexicographic product graphs

A total coloring of a graph G is an assignment of colors to all the elements (vertices and edges) of the graph in such a way that no two adjacent or incident elements receive the same color. The Total Chromatic Number, [Formula: see text] is the minimum number of colors which need to be assigned to obtain a total coloring of the graph [Formula: see text]. The Total Coloring Conjecture made independently by Behzad and Vizing claims that, [Formula: see text], where [Formula: see text] represents the maximum degree of [Formula: see text]. The lower bound is sharp, the upper bound remains to be proved. In this paper, we prove the Total Coloring Conjecture for certain classes of lexicographic product and deleted lexicographic product of graphs.

Injective edge-coloring of graphs with given maximum degree

A coloring of edges of a graph G is injective if for any two distinct edges e1 and e2, the colors of e1 and e2 are distinct if they are at distance 1 in G or in a common triangle. Naturally, the injective chromatic index of G, χinj′(G), is the minimum number of colors needed for an injective edge-coloring of G. We study how large can be the injective chromatic index of G in terms of maximum degree of G when we have restrictions on girth and/or chromatic number of G. We also compare our bounds with analogous bounds on the strong chromatic index.

Digraphs with proper connection number two

A directed path in a digraph is proper if any two consecutive arcs on the path have distinct colors. An arc-colored digraph D is proper connected if for any two distinct vertices x and y of D, there are both proper (x,y)-directed paths and proper (y,x)-directed paths in D. The proper connection number pc→(D) of a digraph D is the minimum number of colors that can be used to make D proper connected. Obviously, if a digraph has a proper connection number, it must be strongly connected, and pc→(D)=1 if and only if D is complete. Magnant et al. showed that pc→(D)≤3 for all strong digraphs D, and Ducoffe et al. proved that deciding whether a given digraph has proper connection number at most two is NP-complete. In this paper, we give a few classes of strong digraphs with proper connection number two, and from our proofs one can construct an optimal arc-coloring for a digraph of order n in time O(n3).

Further Results on the Rainbow Vertex-Disconnection of Graphs

Let G be a nontrivial connected and vertex-colored graph. A subset X of the vertex set of G is called rainbow if any two vertices in X have distinct colors. The graph G is called rainbow vertex-disconnected if for any two vertices x and y of G, there exists a vertex subset S such that when x and y are nonadjacent, S is rainbow and x and y belong to different components of \(G-S\); whereas when x and y are adjacent, \(S+x\) or \(S+y\) is rainbow and x and y belong to different components of \((G-xy)-S\). Such a vertex subset S is called an x–y rainbow vertex-cut of G. For a connected graph G, the rainbow vertex-disconnection number of G, denoted by rvd(G), is the minimum number of colors that are needed to make G rainbow vertex-disconnected. In this paper, we obtain bounds of the rainbow vertex-disconnection number of a graph in terms of the minimum degree and maximum degree of the graph. We give a tighter upper bound for the maximum size of a graph G with \(rvd(G)=k\) for \(k\ge \frac{n}{2}\). We then characterize the graphs of order n with rainbow vertex-disconnection number \(n-1\) and obtain the maximum size of a graph G with \(rvd(G)=n-1\). Moreover, we get a sharp threshold function for the property \(rvd(G(n,p))=n\) and prove that almost all graphs G have \(rvd(G)=rvd({\overline{G}})=n\). Finally, we obtain some Nordhaus–Gaddum-type results: \(n-5\le rvd(G)+rvd({\overline{G}})\le 2n\) and \(n-1\le rvd(G)\cdot rvd({\overline{G}})\le n^2\) for the rainbow vertex-disconnection numbers of nontrivial connected graphs G and \({\overline{G}}\) with order \(n\ge 24\).

Conflict-free Connection Number and Independence Number of a Graph

An edge-colored graph G is conflict-free connected if any two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The conflict-free connection number of a connected graph G, denoted by cfc(G), is defined as the minimum number of colors that are required in order to make G conflict-free connected. In this paper, we investigate the relation between the conflict-free connection number and the independence number of a graph. We firstly show that cfc(G) ≤ α(G) for any connected graph G, and give an example to show that the bound is sharp. With this result, we prove that if T is a tree with \(\Delta(T)\geq\frac{\alpha(T)+2}{2}\), then cfc(T) = Δ(T).

On the rainbow connection number of triangle-net graph

Let G be an arbitrary non-trivial connected graph. For every two vertices u and v in G, a (u,v)-path in G is called a rainbow (u,v)-path if all edges are colored differently. Next, a rainbow (u, v)-geodesic in G is a rainbow (u,v)-path of length d(u,v). Graph G is rainbow connected if for every two vertices u,v ∈ V(G), there exists a rainbow (u,v)-path. If there exists a rainbow (u, v)-geodesic in G for every two vertices u, v ∈ V(G) then G is strongly rainbow connected. The rainbow connection number rc(G) is the minimum number of colors needed to make G rainbow connected, while the strong rainbow connection number src(G) is the minimum number of colors needed to make G strongly rainbow connected. Let Trn be the generalized triangle-ladder graph for n ≥ 2. The triangle-net graph, denoted by H = (Trn)m, is constructed by taking m homogeneous generalized triangle-ladder graphs and identifying their terminal vertices, for m ≥ 2. This paper determined the rainbow connection number of the triangle-net graph and the upper bound of the strong rainbow connection number of the graph.

On the local antimagic labeling of graphs amalgamation

Let G be a simple connected graph, an ordered pair of sets G(V, E), with V is a set of vertices and E is a set of edges. Graph coloring has been one of the most popular branches in topics of graph theory. In 2017 Arumugam et al. developed a new notion of coloring, namely local antimagic coloring of a graph. This concept is a combination of graph labeling and graph coloring. The local antimagic of graphs is one of the colorings of graph theory that is interesting to study. By the definition, the local antimagic coloring is a bijection f : E(G) ƒ> {1, 2, 3,…, |E(G)|} if for any two adjacent vertices u and v, w(u) = w(v), where w(v) = ΣeeE(v) 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 xla(G) is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper, we will study the local antimagic coloring of amalgamation of graphs.

Local antimagic vertex total coloring on fan graph and graph resulting from comb product operation

Let G = (V,E) be a connected graph with |V|=n and |E|=m. A bijection f: V(G) U E(G) → {1,2,3, …,n + m} is called local antimagic vertex total coloring if for any two adjacent vertices u and v, wt(u)≠ wt(v), where wt(u) = ∑e∈E(u) f(e) + f(u), and E (u) is a set of edges incident to u. Thus any local antimagic vertex total labeling induces a proper vertex coloring of G where the vertex v is assigned the color wt (v). The local antimagic vertex total chromatic number χ1να t(G) is the minimum number of colors taken over all colorings induced by local antimagic vertex total. In this paper we investigate local antimagic vertex total coloring on fan graph (Fn) and graph resulting from comb product operation of Fn and F3 which denoted by Fnt> F3. We get two theorems related to the local antimagic vertex total chromatic number. First, χ1να t(Fn) = 3 where n> 3. Second, 3 < χ1ναt(Fn > F3) < 5 where n > 3.