An approach to hedetniemi's conjecture

For a fixed integer n ? ω, a graph G of chromatic number greater than n is called persistent if for all n + 1-chromatic graphs H, the products G × H are n + 1-chromatic graphs. Wheter all graphs of chromatic number greater than n are persistent is a long-standing open problem due to Hedetniemi. We call a graph G strongly persistent if G is persistent and the Hajos sum of G with any other persistent graph H is still persistent. This paper extends the class of known persistent graphs by proving the following results: If G is constructed from copies of K_n+1 by Hajos sums, adding vertices and edges and at most one contraction of nonadjacent vertices, then G is strongly persistent. © 1929 John Wiley & Sons, Inc.     

Partitions of graphs into cographs

Cographs form the minimal family of graphs containing K₁ that is closed with respect to complementation and disjoint union. We discuss vertex partitions of graphs into the smallest number of cographs. We introduce a new parameter, calling the minimum order of such a partition the c-chromatic number of the graph. We begin by axiomatizing several well-known graphical parameters as motivation for this function. We present several bounds on c-chromatic number in terms of well-known expressions. We show that if a graph is triangle-free, then its chromatic number is bounded between the c-chromatic number and twice this number. We show that both bounds are sharp for graphs with arbitrarily high girth. This provides an alternative proof to a result by Broere and Mynhardt; namely, there exist triangle-free graphs with arbitrarily large c-chromatic numbers. We show that any planar graph with girth at least 11 has a c-chromatic number at most two. We close with several remarks on computational complexity. In particular, we show that computing the c-chromatic number is NP-complete for planar graphs.     

The plane‐width of graphs

Map the vertices of a graph to (not necessarily distinct) points of the plane so that two adjacent vertices are mapped at least unit distance apart. The plane‐width of a graph is the minimum diameter of the image of its vertex set over all such mappings. We establish a relation between the plane‐width of a graph and its chromatic number. We also connect it to other well‐known areas, including the circular chromatic number and the problem of packing unit discs in the plane. © 2011 Wiley Periodicals, Inc. J Graph Theory 68: 229‐245, 2011     

Small graphs with chromatic number 5: A computer search

In this article we give examples of a triangle-free graph on 22 vertices with chromatic number 5 and a K₄-free graph on 11 vertices with chromatic number 5. We very briefly describe the computer searches demonstrating that these are the smallest possible such graphs. All 5-critical graphs on 9 vertices are exhibited. © 1995 John Wiley & Sons, Inc.     

Planar unit-distance graphs having planar unit-distance complement

A graph G=(V,E) is called a unit-distance graph in the plane if there is an embedding of V into the plane such that every pair of adjacent vertices are at unit distance apart. If an embedding of V satisfies the condition that two vertices are adjacent if and only if they are at unit distance apart, then G is called a strict unit-distance graph in the plane. A graph G is a (strict) co-unit-distance graph, if both G and its complement are (strict) unit-distance graphs in the plane. We show by an exhaustive enumeration that there are exactly 69 co-unit-distance graphs (65 are strict co-unit-distance graphs), 55 of which are connected (51 are connected strict co-unit-distance graphs), and seven are self-complementary.     

Improved bounds on linear coloring of plane graphs

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest number of colors in a linear coloring of G. In this paper, we give some upper bounds on linear chromatic number for plane graphs with respect to their girth, that improve some results of Raspaud and Wang (2009).     

5-chromatic strongly regular graphs

In this paper, we begin the determination of all primitive strongly regular graphs with chromatic number equal to 5. Using eigenvalue techniques, we show that there are at most 43 possible parameter sets for such a graph. For each parameter set, we must decide which strongly regular graphs, if any, possessing the set are 5-chromatic. In this way, we deal completely with 34 of these parameter sets using eigenvalue techniques and computer enumerations.     

On critical graphs with chromatic index 4

It is shown that the number of vertices of valency 2 in a critical graph with chromatic index 4 is at most 1/3 of the total number of vertices, and that there exist critical graphs with just one vertex of valency 2, but none with exactly two vertices of valency 2. From this bounds for the number of edges are deduced. The paper ends with a presentation of a catalogue of all critical graphs with chromatic index 4 and at most 8 vertices, and all simple critical graphs with chromatic index 4 and at most 10 vertices.     

On some properties of 4-regular plane graphs

The d-distance face chromatic number of a connected plane graph is the minimum number of colors in such a coloring of its faces that whenever two distinct faces are at the distance at most d, they receive distinct colors. We estimate 1-distance chromatic number for connected 4-regular plane graphs. We show that 0-distance face chromatic number of any connected multi-3-gonal 4-regular plane graphs is 4. © 1995, John Wiley & Sons, Inc.     

On the Hardness of Approximating the Chromatic Number

k -colorable for some fixed . Our main result is that it is NP-hard to find a 4-coloring of a 3-chromatic graph. As an immediate corollary we obtain that it is NP-hard to color a k-chromatic graph with at most colors. We also give simple proofs of two results of Lund and Yannakakis [20]. The first result shows that it is NP-hard to approximate the chromatic number to within for some fixed ε > 0. We point here that this hardness result applies only to graphs with large chromatic numbers. The second result shows that for any positive constant h, there exists an integer , such that it is NP-hard to decide whether a given graph G is -chromatic or any coloring of G requires colors. Received April 11, 1997/Revised June 10, 1999     

Lower Bounds for Measurable Chromatic Numbers

The Lovász theta function provides a lower bound for the chromatic number of finite graphs based on the solution of a semidefinite program. In this paper we generalize it so that it gives a lower bound for the measurable chromatic number of distance graphs on compact metric spaces. In particular we consider distance graphs on the unit sphere. There we transform the original infinite semidefinite program into an infinite linear program which then turns out to be an extremal question about Jacobi polynomials which we solve explicitly in the limit. As an application we derive new lower bounds for the measurable chromatic number of the Euclidean space in dimensions 10, . . . , 24 and we give a new proof that it grows exponentially with the dimension.     

On the Chromatic Number of the Visibility Graph of a Set of Points in the Plane

The visibility graph V(P) of a point set P \subseteq R² has vertex set P, such that two points v,w ∈ P are adjacent whenever there is no other point in P on the line segment between v and w. We study the chromatic number of V(P). We characterise the 2- and 3-chromatic visibility graphs. It is an open problem whether the chromatic number of a visibility graph is bounded by its clique number. Our main result is a super-polynomial lower bound on the chromatic number (in terms of the clique number).     

On acyclic colorings of graphs on surfaces

A properk-coloring of a graph is acyclic if every 2-chromatic subgraph is acyclic. Borodin showed that every planar graph has an acyclic 5-coloring. This paper shows that the acyclic chromatic number of the projective plane is at most 7. The acyclic chromatic number of an arbitrary surface with Euler characteristic η=−γ is at mostO(γ^4/7). This is nearly tight; for every γ>0 there are graphs embeddable on surfaces of Euler characteristic −γ whose acyclic chromatic number is at least Ω(γ^4/7/(logγ)^1/7). Therefore, the conjecture of Borodin that the acyclic chromatic number of any surface but the plane is the same as its chromatic number is false for all surfaces with large γ (and may very well be false for all surfaces). This author's research was supported in part by a United States Israeli BSF grant. This author's research was supported by the Ministry of Research and Technology of Slovenia, Research Project P1-0210-101-92. This author's research was supported by the Office of Naval Research, grant number N00014-92-J-1965.     

Coloring, location and domination of corona graphs

A vertex coloring of a graph G is an assignment of colors to the vertices of G so that every two adjacent vertices of G have different colors. A coloring related property of a graphs is also an assignment of colors or labels to the vertices of a graph, in which the process of labeling is done according to an extra condition. A set S of vertices of a graph G is a dominating set in G if every vertex outside of S is adjacent to at least one vertex belonging to S. A domination parameter of G is related to those structures of a graph that satisfy some domination property together with other conditions on the vertices of G. In this article we study several mathematical properties related to coloring, domination and location of corona graphs. We investigate the distance-k colorings of corona graphs. Particularly, we obtain tight bounds for the distance-2 chromatic number and distance-3 chromatic number of corona graphs, through some relationships between the distance-k chromatic number of corona graphs and the distance-k chromatic number of its factors. Moreover, we give the exact value of the distance-k chromatic number of the corona of a path and an arbitrary graph. On the other hand, we obtain bounds for the Roman dominating number and the locating–domination number of corona graphs. We give closed formulaes for the k-domination number, the distance-k domination number, the independence domination number, the domatic number and the idomatic number of corona graphs.     

Δ(G)=5的2-连通外平面图的邻点可区别全染色

Smarandachely邻点可区别全染色是指相邻点的色集合互不包含的邻点可区别全染色，是对邻点可区别全染色条件的进一步加强。本文研究了平面图的Smarandachely邻点可区别全染色，即根据2-连通外平面图的结构特点，利用分析法、数学归纳法，刻画了最大度为5的2-连通外平面图的Smarandachely邻点可区别全色数。证明了：如果$G$是一个$\Delta （G）=5$的2-连通外平面图，则$\chi_{\rm sat}（G）\leqslant 9$。     

具有较小直径的三类图的对极色数

对一个连通图G,令d(u,v)表示G中两个顶点间u和v之间的距离,d表示G的直径.G的一个对极染色指的是从G的顶点集到正整数集(颜色集)的一个映射c,使得对G的任意两个不同的顶点u和v满足d(u,v)+|c(u)-c(v)|≥d.由c映射到G的顶点的最大颜色称为c的值,记作ac(c),而对G的所有对极染色c,ac(c)的最小值称为G的对极色数,记作ac(G).本文确定了轮图、齿轮图以及双星图三类图的对极色数,这些图都具有较小的直径d.     

色数等于选择数的4-正则图

设Hn(n≥5)表示一个图:以1,2,．．．,n为顶点,两个点i和j是相邻的当且仅当|i-j|≤2,其中加法取模n．这篇文章证明了,Hn的色数等于它的选择数．结果被用于刻画最大度至多2的图的列表全色数．     

On the Chromatic Number of \mathbb {R}^4

The lower bound for the chromatic number of \(\mathbb {R}^4\) is improved from \(7\) to \(9\) . Three graphs with unit distance embeddings in \(\mathbb {R}^4\) are described. The first is a \(7\) -chromatic graph of order \(14\) whose chromatic number can be verified by inspection. The second is an \(8\) -chromatic graph of order \(26\) . In this case the chromatic number can be verified quickly by a simple computer program. The third graph is a \(9\) -chromatic graph of order \(65\) for which computer verification takes about one minute.     

Oriented Coloring of Triangle-Free Planar Graphs and 2-Outerplanar Graphs

A graph is planar if it can be embedded on the plane without edge-crossings. A graph is 2-outerplanar if it has a planar embedding such that the subgraph obtained by removing the vertices of the external face is outerplanar (i.e. with all its vertices on the external face). An oriented k-coloring of an oriented graph G is a homomorphism from G to an oriented graph H of order k. We prove that every oriented triangle-free planar graph has an oriented chromatic number at most 40, that improves the previous known bound of 47 [Borodin, O. V. and Ivanova, A. O., An oriented colouring of planar graphs with girth at least 4, Sib. Electron. Math. Reports, vol. 2, 239–249, 2005]. We also prove that every oriented 2-outerplanar graph has an oriented chromatic number at most 40, that improves the previous known bound of 67 [Esperet, L. and Ochem, P. Oriented colouring of 2-outerplanar graphs, Inform. Process. Lett., vol. 101(5), 215–219, 2007].     

List 2-facial 5-colorability of plane graphs with girth at least 12

A proper vertex coloring of a plane graph is 2-facial if any two different vertices joined by a facial walk of length 2 are colored differently, and it is 2-distance if every two vertices at distance 2 from each other are colored differently. Note that any 2-facial coloring of a subcubic graph is 2-distance.It is known that every plane graph with girth at least 14 has a 2-facial 5-coloring [M. Montassier, A. Raspaud, A note on 2-facial coloring of plane graphs. Inform. Process. Lett. 98 (6) (2006) 235–241], and that every planar subcubic graph with girth at least 13 has a list 2-distance 5-coloring [F. Havet, Choosability of square of planar subcubic graphs with large girth, Discrete Math. 309 (2009) 3353–3563].We strengthen these results by proving the list 2-facial 5-colorability of plane graphs with girth at least 12.