The structure of plane graphs with independent crossings and its applications to coloring problems

If a graph G has a drawing in the plane in such a way that every two crossings are independent, then we call G a plane graph with independent crossings or IC-planar graph for short. In this paper, the structure of IC-planar graphs with minimum degree at least two or three is studied. By applying their structural results, we prove that the edge chromatic number of G is Δ if Δ ≥ 8, the list edge (resp. list total) chromatic number of G is Δ (resp. Δ + 1) if Δ ≥ 14 and the linear arboricity of G is ?Δ/2? if Δ ≥ 17, where G is an IC-planar graph and Δ is the maximum degree of G.     

Decompositions for edge-coloring join graphs and cobipartite graphs

An edge-coloring is an association of colors to the edges of a graph, in such a way that no pair of adjacent edges receive the same color. A graph G is Class 1 if it is edge-colorable with a number of colors equal to its maximum degree Δ(G). To determine whether a graph G is Class 1 is NP-complete [I. Holyer, The NP-completeness of edge-coloring, SIAM J. Comput. 10 (1981) 718-720]. First, we propose edge-decompositions of a graph G with the goal of edge-coloring G with Δ(G) colors. Second, we apply these decompositions for identifying new subsets of Class 1 join graphs and cobipartite graphs. Third, the proposed technique is applied for proving that the chromatic index of a graph is equal to the chromatic index of its semi-core, the subgraph induced by the maximum degree vertices and their neighbors. Finally, we apply these decomposition tools to a classical result [A.J.W. Hilton, Z. Cheng, The chromatic index of a graph whose core has maximum degree 2, Discrete Math. 101 (1992) 135-147] that relates the chromatic index of a graph to its core, the subgraph induced by the maximum degree vertices.     

Planar graphs with maximum degree 8 and without intersecting chordal 4-cycles are 9-totally colorable

The minimum number of colors needed to properly color the vertices and edges of a graph G is called the total chromatic number of G and denoted by χ’’ (G). It is shown that if a planar graph G has maximum degree Δ≥9, then χ’’ (G) = Δ + 1. In this paper, we prove that if G is a planar graph with maximum degree 8 and without intersecting chordal 4-cycles, then χ ’’(G) = 9.     

Total coloring of graphs embedded in surfaces of nonnegative Euler characteristic

Let G be a graph which can be embedded in a surface of nonnegative Euler characteristic.In this paper,it is proved that the total chromatic number of G is △(G)+1 if △(G)9,where △(G)is the maximum degree of G.     

Total coloring of embedded graphs of maximum degree at least ten

A total k-coloring of a graph G is a coloring of V(G) ∪ E(G) using k colors such that no two adjacent or incident elements receive the same color.The total chromatic number χ〃(G) is the smallest integer k such that G has a total k-coloring.In this paper,it is proved that the total chromatic number of any graph G embedded in a surface Σ of Euler characteristic χ(Σ)≥0 is Δ(G) + 1 if Δ(G)≥10,where Δ(G) denotes the maximum degree of G.     

Game chromatic number of strong product graphs

The graph coloring game is a two-player game in which the two players properly color an uncolored vertex of G alternately. The first player wins the game if all vertices of G are colored, and the second wins otherwise. The game chromatic number of a graph G is the minimum integer k such that the first player has a winning strategy for the graph coloring game on G with k colors. There is a lot of literature on the game chromatic number of graph products, e.g., the Cartesian product and the lexicographic product. In this paper, we investigate the game chromatic number of the strong product of graphs, which is one of major graph products. In particular, we completely determine the game chromatic number of the strong product of a double star and a complete graph. Moreover, we estimate the game chromatic number of some King's graphs, which are the strong products of two paths.     

Acyclic edge coloring of graphs with large girths

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G, denoted by χ’a(G), is the least number of colors such that G has an acyclic edge k-coloring. Let G be a graph with maximum degree Δ and girth g(G), and let 1≤r≤2Δ be an integer. In this paper, it is shown that there exists a constant c > 0 such that if g(G)≥cΔ r log(Δ2/r) then χa(G)≤Δ + r + 1, which generalizes the result of Alon et al. in 2001. When G is restricted to series-parallel graphs, it is proved that χ’a(G) = Δ if Δ≥4 and g(G)≥4; or Δ≥3 and g(G)≥5.     

The dichromatic number of a digraph

In this paper the concept of dichromatic number of a digraph which is a generalization of the chromatic number of a graph is introduced. The dichromatic number of a digraph D is defined as the minimum number of colours required to colour the vertices of D in such a way that the chromatic classes induce acyclic subdigraphs in D. Some results relating the dichromatic number of D with the existence of cycles of special lengths in D are presented. Contributions to chromatic theory are also obtained. In particular, we generalize the theorem due to P. Erdös and A. Hajnal (Acta Math. Acad. Sci. Hungar.17 (1966), 61–99) which states the existence of odd cycles of length ≥_χ(G) ? 1 in any graph G.     

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).     

Acyclic improper colouring of graphs with maximum degree 4

A k-colouring(not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We consider acyclic k-colourings such that each colour class induces a graph with a given(hereditary) property. In particular, we consider acyclic k-colourings in which each colour class induces a graph with maximum degree at most t, which are referred to as acyclic t-improper k-colourings. The acyclic t-improper chromatic number of a graph G is the smallest k for which there exists an acyclic t-improper k-colouring of G. We focus on acyclic colourings of graphs with maximum degree 4. We prove that 3 is an upper bound for the acyclic 3-improper chromatic number of this class of graphs. We also provide a non-trivial family of graphs with maximum degree4 whose acyclic 3-improper chromatic number is at most 2, namely, the graphs with maximum average degree at most 3. Finally, we prove that any graph G with Δ(G) 4 can be acyclically coloured with 4 colours in such a way that each colour class induces an acyclic graph with maximum degree at most 3.     

Oriented coloring of triangle-free planar graphs and 2-outerplanar graphs

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 (1) every oriented triangle-free planar graph has an oriented chromatic number at most 40, and (2) every oriented 2-outerplanar graph has an oriented chromatic number at most 40, that improves the previous known bounds of 47 and 67, respectively.     

Chromatically Supremal Decompositions of Graphs

If G is a graph, a G-decomposition of a host graph H is a partition of the edges of H into subgraphs of H which are isomorphic to G. The chromatic index of a G-decomposition of H is the minimum number of colors required to color the parts of the decomposition so that parts which share a common node get different colors. We establish an upper bound on the chromatic index and characterize those decompositions which achieve it. The structurally most interesting of the decompositions with maximal chromatic index are associated with (v, k, 1)-designs.     

Even polyhedral decompositions of cubic graphs

An even polyhedral decomposition of a finite cubic graph G is defined as a set of elementary cycles of even length in G with the property that each edge of G lies in exactly two of them. If G has chromatic index three, then G has an even polyhedral decomposition. We show that, contrary to a theorem of Szekkeres [2], this property to have an even polyhedral decomposition doesn't characterize the cubic graphs of chromatic index three. In particular, there exists an infinite family of sharks all having an even polyhedral decomposition.     

On chromatic‐choosable graphs

A graph is chromatic‐choosable if its choice number coincides with its chromatic number. It is shown in this article that, for any graph G, if we join a sufficiently large complete graph to G, then we obtain a chromatic‐choosable graph. As a consequence, if the chromatic number of a graph G is close enough to the number of vertices in G, then G is chromatic‐choosable. We also propose a conjecture related to this fact. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 130–135, 2002     

The book thickness of a graph

The book thickness bt(G) of a graph G is defined, its basic properties are delineated, and relations are given with other invariants such as thickness, genus, and chromatic number. A graph G has book thickness bt(G) ≤ 2 if and only if it is a subgraph of a hamiltonian planar graph, but we conjecture that there are planar graphs with arbitrarily high book thickness.     

Optimal acyclic edge‐coloring of cubic graphs

An acyclic edge‐coloring of a graph is a proper edge‐coloring such that the subgraph induced by the edges of any two colors is acyclic. The acyclic chromatic index of a graph G is the smallest number of colors in an acyclic edge‐coloring of G. We prove that the acyclic chromatic index of a connected cubic graph G is 4, unless G is K₄ or K_3,3; the acyclic chromatic index of K₄ and K_3,3 is 5. This result has previously been published by Fiam?ík, but his published proof was erroneous.     

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.     

Backbone colorings of graphs with bounded degree

We study backbone colorings, a variation on classical vertex colorings: Given a graph G and a subgraph H of G (the backbone of G), a backbone coloring for G and H is a proper vertex k-coloring of G in which the colors assigned to adjacent vertices in H differ by at least 2. The minimal k∈N for which such a coloring exists is called the backbone chromatic number of G. We show that for a graph G of maximum degree Δ where the backbone graph is a d-degenerated subgraph of G, the backbone chromatic number is at most Δ+d+1 and moreover, in the case when the backbone graph being a matching we prove that the backbone chromatic number is at most Δ+1. We also present examples where these bounds are attained.Finally, the asymptotic behavior of the backbone chromatic number is studied regarding the degrees of G and H. We prove for any sparse graph G that if the maximum degree of a backbone graph is small compared to the maximum degree of G, then the backbone chromatic number is at most .     

Improved Upper Bound for Generalized Acyclic Chromatic Number of Graphs

A vertex coloring of a graph G is called r-acyclic if it is a proper vertex coloring such that every cycle D receives at least min{|D|, r} colors. The r-acyclic chromatic number of G is the least number of colors in an r-acyclic coloring of G. We prove that for any number r ≥ 4, the r-acyclic chromatic number of any graph G with maximum degree Δ ≥ 7 and with girth at least (r ? 1)Δ is at most (4r ? 3)Δ.