Star Chromatic Number Research Articles

Star chromatic number of some graph products

A star coloring of a graph [Formula: see text] is a proper vertex coloring in which every path on four vertices uses at least three distinct colors. Equivalently, in a star coloring, the induced subgraphs formed by the vertices of any two colors have connected components that are star graphs. A graph [Formula: see text] is [Formula: see text]- star-colorable if there exists a star coloring of [Formula: see text] from a set of [Formula: see text] colors. The minimum positive integer [Formula: see text] for which [Formula: see text] is [Formula: see text]-star-colorable is the star chromatic number of [Formula: see text] and is denoted by [Formula: see text]. In this paper, upper and lower bounds are presented for the star chromatic number of the rooted product, hierarchical product, and lexicographic product.

Star colouring of bounded degree graphs and regular graphs

A k-star colouring of a graph G is a function f:V(G)→{0,1,…,k−1} such that f(u)≠f(v) for every edge uv of G, and every bicoloured connected subgraph of G is a star. The star chromatic number of G, χs(G), is the least integer k such that G is k-star colourable. We prove that χs(G)≥⌈(d+4)/2⌉ for every d-regular graph G with d≥3. We reveal the structure and properties of even-degree regular graphs G that attain this lower bound. The structure of such graphs G is linked with a certain type of Eulerian orientations of G. Moreover, this structure can be expressed in the LC-VSP framework of Telle and Proskurowski (SIDMA, 1997), and hence can be tested by an FPT algorithm with the parameter either treewidth, cliquewidth, or rankwidth. We prove that for p≥2, a 2p-regular graph G is (p+2)-star colourable only if n:=|V(G)| is divisible by (p+1)(p+2). For each p≥2 and n divisible by (p+1)(p+2), we construct a 2p-regular Hamiltonian graph on n vertices which is (p+2)-star colourable.The problem k-Star Colourability takes a graph G as input and asks whether G is k-star colourable. We prove that 3-Star Colourability is NP-complete for planar bipartite graphs of maximum degree three and arbitrarily large girth. Besides, it is coNP-hard to test whether a bipartite graph of maximum degree eight has a unique 3-star colouring up to colour swaps. For k≥3, k-Star Colourability of bipartite graphs of maximum degree k is NP-complete, and does not even admit a 2o(n)-time algorithm unless ETH fails.

On star and acyclic coloring of generalized lexicographic product of graphs

<abstract><p>A $ star \; coloring $ of a graph $ G $ is a proper vertex coloring of $ G $ such that any path of length 3 in $ G $ is not bicolored. The $ star \; chromatic \; number $ $ \chi_s(G) $ of $ G $ is the smallest integer $ k $ for which $ G $ admits a star coloring with $ k $ colors. A $ acyclic \; coloring $ of $ G $ is a proper coloring of $ G $ such that any cycle in $ G $ is not bicolored. The $ acyclic \; chromatic \; number $ of $ G $, denoted by $ a(G) $, is the minimum number of colors needed to acyclically color $ G $. In this paper, we present upper bound for the star and acyclic chromatic numbers of the generalized lexicographic product $ G[h_n] $ of graph $ G $ and disjoint graph sequence $ h_n $, where $ G $ exists a $ k- $colorful neighbor star coloring or $ k- $colorful neighbor acyclic coloring. In addition, the upper bounds are tight.</p></abstract>

Star coloring under some graph operations

A star coloring of a graph [Formula: see text] is a proper coloring of [Formula: see text] such that no path of length [Formula: see text] in [Formula: see text] is bicolored. In this paper, the star chromatic number of join of graphs is computed. Some sharp bounds for the star chromatic number of corona, lexicographic, deleted lexicographic and hierarchical product of graphs together with a conjecture on the star chromatic number of lexicographic product of graphs are also presented.

From χ- to χp-bounded classes

χ-bounded classes are studied here in the context of star colorings and, more generally, χp-colorings. This fits to a general scheme of sparsity and leads to natural extensions of the notion of bounded expansion class. In this paper we solve two conjectures related to star coloring (i.e. χ2) boundedness. One of the conjectures is disproved and in fact we determine which weakening holds true. χp-boundedness leads to more stability and we give structural characterizations of (strong and weak) χp-bounded classes. We also generalize a result of Wood relating the chromatic number of a graph to the star chromatic number of its 1-subdivision. As an application of our characterizations, among other things, we show that for every odd integer g>3 even hole-free graphs G contain at most φ(g,ω(G))|G| holes of length g.

On star 5-colorings of sparse graphs

A stark-coloring of a graph G is a proper (vertex) k-coloring of G such that the vertices on a path of length three receive at least three colors. Given a graph G, its star chromatic number, denoted χs(G), is the minimum integer k for which G admits a star k-coloring. Studying star coloring of sparse graphs is an active area of research, especially in terms of the maximum average degree of a graph; the maximum average degree, denoted mad(G), of a graph G is max2|E(H)||V(H)|:H⊂G. It is known that for a graph G, if mad(G)<83, then χs(G)≤6 (Kündgen and Timmons, 2010), and if mad(G)<187 and its girth is at least 6, then χs(G)≤5 (Bu et al., 2009). We improve both results by showing that for a graph G, if mad(G)≤83, then χs(G)≤5. As an immediate corollary, we obtain that a planar graph with girth at least 8 has a star 5-coloring, improving the best known girth condition for a planar graph to have a star 5-coloring (Kündgen and Timmons, 2010; Timmons, 2008).

Star chromatic number of some graphs

A proper vertex coloring of a graph [Formula: see text] is called a star coloring if every two color classes induce a forest whose each component is a star, which means there is no bicolored [Formula: see text] in [Formula: see text]. In this paper, we show that the Cartesian product of any two cycles, except [Formula: see text] and [Formula: see text], has a [Formula: see text]-star coloring.

Star chromatic number of Harary graphs

A proper vertex coloring of a graph [Formula: see text] is called a star coloring if every path on four vertices in [Formula: see text] is not 2-colored. The star chromatic number is the minimum number of colors required to star color [Formula: see text] and it is denoted by [Formula: see text]. In this paper, the star coloring of Harary graphs [Formula: see text], where [Formula: see text] is even and [Formula: see text] is odd, is discussed.

On star coloring of modular product of graphs

A star coloring of a graph $G$ is a proper vertex coloring in which every path on four vertices in $G$ is not bicolored. The star chromatic number $\chi_{s}\left(G\right)$ of $G$ is the least number of colors needed to star color $G$. In this paper, we find the exact values of the star chromatic number of modular product of complete graph with complete graph $K_m \diamond K_n$, path with complete graph $P_m \diamond K_n$ and star graph with complete graph $K_{1,m}\diamond K_n$. \par All graphs in this paper are finite, simple, connected and undirected graph and we follow \cite{bm, cla, f} for terminology and notation that are not defined here. We denote the vertex set and the edge set of $G$ by $V(G)$ and $E(G)$, respectively. Branko Gr\"{u}nbaum introduced the concept of star chromatic number in 1973. A star coloring \cite{alberton, fertin, bg} of a graph $G$ is a proper vertex coloring in which every path on four vertices uses at least three distinct colors. The star chromatic number $\chi_{s}\left(G\right)$ of $G$ is the least number of colors needed to star color $G$. \par During the years star coloring of graphs has been studied extensively by several authors, for instance see \cite{alberton, col, fertin}.

ON STAR COLORING OF DEGREE SPLITTING OF COMB PRODUCT GRAPHS

A star coloring of a graph G is a proper vertex coloring in which every path on four vertices in G is not bicolored. The star chromatic number χs (G) of G is the least number of colors needed to star color G. Let G = (V,E) be a graph with V = S1 [ S2 [ S3 [ . . . [ St [ T where each Si is a set of all vertices of the same degree with at least two elements and T =V (G) − St i=1 Si. The degree splitting graph DS (G) is obtained by adding vertices w1,w2, . . .wt and joining wi to each vertex of Si for 1 i t. The comb product between two graphs G and H, denoted by G ⊲ H, is a graph obtained by taking one copy of G and |V (G)| copies of H and grafting the ith copy of H at the vertex o to the ith vertex of G. In this paper, we give the exact value of star chromatic number of degree splitting of comb product of complete graph with complete graph, complete graph with path, complete graph with cycle, complete graph with star graph, cycle with complete graph, path with complete graph and cycle with path graph.

On star coloring of degree splitting of join graphs

A star coloring of a graph G is a proper vertex coloring in which every path on four vertices in G is not bicolored. The star chromatic number χs (G) of G is the least number of colors needed to star color G. In this paper, we have generalized the star chromatic number of degree splitting of join of any two graph G and H denoted by G + H, where G is a path graph and H is any simple graph. Also, we determine the star chromatic number for degree splitting of join of path graph G of order m with path Pn, complete graph Kn and cyclevgraph Cn.

Edge, acyclic and star colorings on human chain graphs

In this paper, we determine chromatic index, acyclic chromatic and star chromatic number for the human chain and circular human chain graphs.

The star dichromatic number

We introduce a new notion of circular colourings for digraphs. The idea of this quantity, called star dichromatic number $\vec{\chi}^\ast(D)$ of a digraph $D$, is to allow a finer subdivision of digraphs with the same dichromatic number into such which are easier or harder to colour by allowing fractional values. This is related to a coherent notion for the vertex arboricity of graphs introduced by Wang et al. and resembles the concept of the star chromatic number of graphs introduced by Vince in the framework of digraph colouring. After presenting basic properties of the new quantity, including range, simple classes of digraphs, general inequalities and its relation to integer counterparts as well as other concepts of fractional colouring, we compare our notion with the notion of circular colourings for digraphs introduced by Bokal et al. and point out similarities as well as differences in certain situations. As it turns out, the star dichromatic number is a lower bound for the circular dichromatic number of Bokal et al., but the gap between the numbers may be arbitrarily close to $1$. We conclude with a discussion of the case of planar digraphs and point out some open problems.

Graph 2-rankings

A 2-ranking of a graph G is an ordered partition of the vertices of G into independent sets $$V_1, \ldots , V_t$$ such that for $$i<j$$ , the subgraph of G induced by $$V_i \cup V_j$$ is a star forest in which each vertex in $$V_i$$ has degree at most 1. A 2-ranking is intermediate in strength between a star coloring and a distance-2 coloring. The 2-ranking number ofG, denoted $$\chi _{2}(G)$$ , is the minimum number of parts needed for a 2-ranking. For the d-dimensional cube $$Q_d$$ , we prove that $$\chi _{2}(Q_d) = d+1$$ . As a corollary, we improve the upper bound on the star chromatic number of products of cycles when each cycle has length divisible by 4. Let $$\chi _{2}'(G)=\chi _{2}(L(G))$$ , where L(G) is the line graph of G; equivalently, $$\chi _{2}'(G)$$ is the minimum t such that there is an ordered partition of E(G) into t matchings $$M_1, \ldots , M_t$$ such that for each j, the matching $$M_j$$ is induced in the subgraph of G with edge set $$M_1 \cup \cdots \cup M_j$$ . We show that $$\chi _{2}'(K_{m,n})=nH_m$$ when m! divides n, where $$K_{m,n}$$ is the complete bipartite graph with parts of sizes m and n, and $$H_m$$ is the harmonic sum $$1 + \cdots + \frac{1}{m}$$ . We also prove that $$\chi _{2}(G) \le 7$$ when G is subcubic and show the existence of a graph G with maximum degree k and $$\chi _{2}(G) \ge \varOmega (k^2/\log (k))$$ .

ON STAR COLOURING OF M(TM,N),M(TN),M(LN) AND M(SN)

A Star coloring of an undirected graph G is a proper vertex coloring of G in which every path on four vertices contains at least three distinct colors. The Star chromatic number of an undirected graph Χs(G), denoted by(G) is the smallest integer k for which G admits a star coloring with k colors. In this paper, we obtain the exact value of the Star chromatic number of Middle graph of Tadpole graph, Snake graph, Ladder graph and Sunlet graphs denoted by M(Tm,n), M(Tn),M(Ln) and M(Sn) respectively.

On star coloring of Mycielskians

&lt;p&gt;In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph &lt;span class="math"&gt;&lt;em&gt;G&lt;/em&gt;&lt;/span&gt; into a new graph &lt;span class="math"&gt;&lt;em&gt;μ&lt;/em&gt;(&lt;em&gt;G&lt;/em&gt;)&lt;/span&gt;, we now call the Mycielskian of &lt;span class="math"&gt;&lt;em&gt;G&lt;/em&gt;&lt;/span&gt;, which has the same clique number as &lt;span class="math"&gt;&lt;em&gt;G&lt;/em&gt;&lt;/span&gt; and whose chromatic number equals &lt;span class="math"&gt;&lt;em&gt;χ&lt;/em&gt;(&lt;em&gt;G&lt;/em&gt;) + 1&lt;/span&gt;. In this paper, we find the star chromatic number for the Mycielskian graph of complete graphs, paths, cycles and complete bipartite graphs.&lt;/p&gt;

Characterization of forbidden subgraphs for bounded star chromatic number

The chromatic number of a graph is the minimum k such that the graph has a proper k-coloring. There are many coloring parameters in the literature that are proper colorings that also forbid bicolored subgraphs. Some examples are 2-distance coloring, acyclic coloring, and star coloring, which forbid a bicolored path on three vertices, bicolored cycles, and a bicolored path on four vertices, respectively. This notion was first suggested by Grünbaum in 1973, but no specific name was given. We revive this notion by defining an H-avoidingk-coloring to be a proper k-coloring that forbids a bicolored subgraph H.When considering the class C of graphs with no F as an induced subgraph, it is not hard to see that every graph in C has bounded chromatic number if and only if F is a complete graph of size at most two. We study this phenomena for the class of graphs with no F as a subgraph for H-avoiding coloring. We completely characterize all graphs F where the class of graphs with no F as a subgraph has bounded H-avoiding chromatic number for a large class of graphs H. As a corollary, our main result implies a characterization of graphs F where the class of graphs with no F as a subgraph has bounded star chromatic number. We also obtain a complete characterization for the acyclic chromatic number.

Improved upper bound for the degenerate and star chromatic numbers of graphs

Let \(G=G(V,E)\) be a graph. A proper coloring of G is a function \(f:V\rightarrow N\) such that \(f(x)\ne f(y)\) for every edge \(xy\in E\). A proper coloring of a graph G such that for every \(k\ge 1\), the union of any k color classes induces a \((k-1)\)-degenerate subgraph is called a degenerate coloring; a proper coloring of a graph with no two-colored \(P_{4}\) is called a star coloring. If a coloring is both degenerate and star, then we call it a degenerate star coloring of graph. The corresponding chromatic number is denoted as \(\chi _{sd}(G)\). In this paper, we employ entropy compression method to obtain a new upper bound \(\chi _{sd}(G)\le \lceil \frac{19}{6}\Delta ^{\frac{3}{2}}+5\Delta \rceil \) for general graph G.

Star chromatic bounds

A star coloring of an undirected graph G is a coloring of the vertices of G such that (i) no two adjacent vertices receive the same color, and (ii) no path on four vertices (not necessarily induced) is bi-colored. The star chromatic number of G is the minimum number of colors needed to star color G. In this note, we deduce upper bounds for the star chromatic number in terms of the clique number for some special classes of graphs which are defined by forbidden induced subgraphs.

Star coloring of cubic graphs

A star coloring of a graph G is a proper vertex coloring of G such that any path of length 3 in G is not bicolored. The star chromatic number of a graph G, denoted by χs(G), is the smallest integer k for which G admits a star coloring with k colors. In this paper, we show that if G is a cubic graph, then χs(G)≥4 and the lower bound is optimal.