Oriented colorings of 2-outerplanar graphs

A graph G is 2-outerplanar if it has a planar embedding such that the subgraph obtained by removing the vertices of the outer face is outerplanar. The oriented chromatic number of an oriented graph H is defined as the minimum order of an oriented graph H^′ such that H has a homomorphism to H^′. In this paper, we prove that 2-outerplanar graphs are 4-degenerate. We also show that oriented 2-outerplanar graphs have a homomorphism to the Paley tournament QR₆₇, which implies that their (strong) oriented chromatic number is at most 67.     

The oriented chromatic number of Halin graphs

The oriented chromatic number of an oriented graph G is the minimum order of an oriented graph H such that G admits a homomorphism to H. The oriented chromatic number of an unoriented graph G is the maximal chromatic number over all possible orientations of G. In this paper, we prove that every Halin graph has oriented chromatic number at most 8, improving a previous bound by Hosseini Dolama and Sopena, and confirming the conjecture given by Vignal.     

On edge covering colorings of graphs

An edge covering coloring of a graph G is an edge-coloring of G such that each color appears at each vertex at least one time. The maximum integer k such that G has an edge covering coloring with k colors is called the edge covering chromatic index of G and denoted by . It is known that for any graph G with minimum degree δ(G), it holds that . Based on the subgraph of G induced by the vertices of minimum degree, we find a new sufficient condition for a graph G to satisfy . This result substantially extends a result of Wang et al. in 2006.     

On edge colorings of 1-planar graphs

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is shown that every 1-planar graph with maximum degree Δ?10 can be edge-colored with Δ colors.     

Minimum cycle bases of weighted outerplanar graphs

We give the first optimal algorithm that computes a minimum cycle basis for any weighted outerplanar graph. Specifically, for any n-node edge-weighted outerplanar graph G, we give an O(n)-time algorithm to obtain an O(n)-space compact representation Z(C) for a minimum cycle basis C of G. Each cycle in C can be computed from Z(C) in O(1) time per edge. Our result works for directed and undirected outerplanar graphs G.     

A note on total colorings of 1-planar graphs

A 1-planar graph is a graph that can be drawn in the plane such that each edge is crossed by at most one other edge. In this paper we give an upper bound for the total chromatic number for 1-planar graphs with maximum degree at least 10.     

The 2-dipath chromatic number of Halin graphs

A 2-dipath k-coloring f of an oriented graph is a mapping from to the color set {1,2,…,k} such that f(x)≠f(y) whenever two vertices x and y are linked by a directed path of length 1 or 2. The 2-dipath chromatic number of is the smallest k such that has a 2-dipath k-coloring. In this paper we prove that if is an oriented Halin graph, then . There exist infinitely many oriented Halin graphs such that .     

Kolmogorov random graphs only have trivial stable colorings

In this paper, we prove that random graphs only have trivial stable colorings. Our result improves Theorem 4.1 in [Proc. 20th IEEE Symp. on Foundations of Comput. Sci., 1979, pp. 39-46]. It can be viewed as an effective version of Corollary 2.13 in [SIAM J. Comput. 29 (2) (2000) 590-599]. As a byproduct, we also give an upper bound of the size of induced regular subgraphs in random graphs.     

List edge and list total colorings of planar graphs without short cycles

Let G be a planar graph with maximum degree Δ(G). We use and to denote the list edge chromatic number and list total chromatic number of G, respectively. In this paper, it is proved that and if Δ(G)?6 and G has neither C₄ nor C₆, or Δ(G)?7 and G has neither C₅ nor C₆, where Ck is a cycle of length k.     

There exist oriented planar graphs with oriented chromatic number at least sixteen

We prove that there exist oriented planar graphs with oriented chromatic number at least 16. Using a result of Raspaud and Sopena [Inform. Process. Lett. 51 (1994) 171-174], this gives that the oriented chromatic number of the family of oriented planar graphs lies between 16 and 80.     

On the oriented chromatic number of Halin graphs

An oriented k-coloring of an oriented graph G is a mapping such that (i) if xy∈E(G) then c(x)≠c(y) and (ii) if xy,zt∈E(G) then c(x)=c(t)⇒c(y)≠c(z). The oriented chromatic number of an oriented graph G is defined as the smallest k such that G admits an oriented k-coloring. We prove in this paper that every Halin graph has oriented chromatic number at most 9, improving a previous bound proposed by Vignal.     

Game chromatic number of graphs with locally bounded number of cycles

We consider the coloring game and the marking game on graphs with bounded number of cycles passing through any edge. We prove that the game coloring number of a graph G is at most c+4, if every edge of G belongs to at most c different cycles. This result covers two earlier bounds on the game coloring number: for trees (c=0) and for cactuses (c=1).     

Some results on acyclic edge coloring of plane graphs

An acyclic edge coloring of a graph is a proper edge coloring without bichromatic cycles. The acyclic chromatic index of a graph G, denoted by α^′(G), is the minimum number k such that G admits an acyclic edge coloring using k colors. Let G be a plane graph with maximum degree Δ and girth g. In this paper, we prove that α^′(G)=Δ(G) if one of the following conditions holds: (1) Δ?8 and g?7; (2) Δ?6 and g?8; (3) Δ?5 and g?9; (4) Δ?4 and g?10; (5) Δ?3 and g?14. We also improve slightly a result of A. Fiedorowicz et al. (2008) [7] by showing that every triangle-free plane graph admits an acyclic edge coloring using at most Δ(G)+5 colors.     

On the adjacent vertex distinguishing edge colourings of graphs

A k-adjacent vertex distinguishing edge colouring or a k-avd-colouring of a graph G is a proper k-edge colouring of G such that no pair of adjacent vertices meets the same set of colours. The avd-chromatic number, denoted by χ′_a(G), is the minimum number of colours needed in an avd-colouring of G. It is proved that for any connected 3-colourable Hamiltonian graph G, we have χ′_a(G)≤Δ+3.