The example of the complete graph k 6, which is 1planar, shows that 1planar graphs may sometimes require six colors. Dualizing, we immediately see that planar triangulations are also vertexface 6choosable. E c where c is the set of available colors with ce 6 cf for any adjacent edges e,f. The techniques used for these two results are the kernel method, the quantitative combinatorial nullstellensatz, and the discharging method. Pdf acyclic edge coloring of planar graphs with colors. An edgecoloring of a graph is a function c from the edges of a graph to a set of colors having the prop erty that if two edges share a common vertex as an. We often write uv to denote an edge with endpoints u 19 and v, even at the risk of confusing the reader when there is more than one such edge. Strong edgecoloring of planar graphs, discrete math. The study of graph colorings has historically been linked closely to that of planar graphs and the four color theorem, which is also the most famous graph coloring problem.
In this paper, we first give a useful structural theorem for 1 planar graphs, and then apply it to the list edge and list total coloring, the p, 1total labelling, and the equitable edge coloring of 1 planar graphs. A strong edgecoloring of a graph is a proper edgecoloring where the edges at distance at most 2 receive distinct colors. May 24, 2017 an edge coloring of a graph g is equitable if, for each vertex v of g, the number of edges of any one color incident with v differs from the number of edges of any other color incident with v by at most one. A graph is planar if a planar embedding of it exists. On the equitable edgecoloring of 1planar graphs and. If a 1 planar graph, one of the most natural generalizations of planar graphs, is drawn that way, the drawing is called a 1plane graph or 1. More precisely, we verify the wellknown list edge coloring conjecture.
Strong edgecoloring of planar graphs sciencedirect. Note that although typically an edge lies in the boundary of two faces, an edge can be in the boundary of a single face, which happens if and only if the edge is a cutedge. Graph coloring and chromatic numbers brilliant math. First, in section 2, we show that every planar graph. Acyclic edge coloring, acyclic edge chromatic number, planar graphs. Acyclic edge coloring of planar graphs with colors. An acyclic coloring of a graph gis a proper vertex coloring of gsuch that the subgraph induced by any two color classes is acyclic. We also find necessary conditions for maximum degree to extend a listedgeprecoloring to eg for a planar graph g. The smallest number of colors for which a strong edgecoloring of a graph g exists is called the strong chromatic index. In this paper, we give some evidences to this conjecture. Important note a graph may be planar even if it is drawn with crossings, because it may be possible to draw it in a different way without crossings.
In topological graph theory, a 1 planar graph is a graph that can be drawn in the euclidean plane in such a way that each edge has at most one crossing point, where it crosses a single additional edge. Now consider a plane drawing of a trianglefree planar graph g on n vertices having the maximum number of edges. The complexity of edgecoloring planar graphs has been considered already in some articles. A graph is 1 planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. We usedgx, or simply dx, to denote the degree of a vertexx in g. Planar graph examples which families of graphs are obviously planar. It is known that every planar graph g has a strong edgecoloring with at most 4. The minimum number of colors required for strong edge. Pdf a strong edgecoloring of a graph is a proper edgecoloring where the edges at distance at most two receive distinct colors. A planar graph is a graph that can be drawn on the plane such that its edges only intersect at their endpoints.
The rst mono graph by fiorini and wilson 41 appeared in 1977 and deals mainly with edge coloring of simple graphs. Main results and their proofs in the section, we always assume that all graphs are planar graphs that have been embedded in the plane andg is a planar graph without 6cycles with two chords. Remember that two vertices are adjacent if they are directly connected by an edge. Npcompleteness of list coloring and precoloring extension. Both edgecoloring and listedgecoloring planar graphs are somewhat simpler. This algorithm is based on a modified proof of vizings theorem for planar graphs.
A planar graph divides the plans into one or more regions. In this paper, we study simple planar graphs which need only. Pdf strong edgecoloring of planar graphs roman sotak. Our results show that the more general list edge coloring and edge precoloring extension problems.
It is known that every planar graph with maximum degree d has a strong. An \emphacyclic edgecoloring of a graph g is a proper edgecoloring of g such that the subgraph induced by any two color classes. However, on the right we have a different drawing of the same graph, which is a plane graph. In this paper, we propose to study homomorphisms of 2edgecolored graphs and the related notion of coloring. In the paper, we prove that every 1planar graph has an equitable edgecoloring with k colors for any integer \k\ge 21\, and every planar graph has an equitable edgecoloring with k.
In this paper, we first give a useful structural theorem for 1planar graphs, and then apply it to the list edge and list total coloring, the p, 1total labelling, and the equitable edge coloring of 1planar graphs. Acyclic edge coloring of triangle free planar graphs. Strong edge coloring of planar graphs with girth at least six. The edgecoloring problem is one of the fundamental prob lems on graphs, which often appears in various scheduling problems like. An edgecoloring of a graph g is equitable if, for each vertex v of g, the number of edges of any one color incident with v differs from the number of edges of any other color incident with v by at most one. Trianglefree planar graphs theorem if g is a trianglefree planar graph with n. In this thesis we prove that triangulations of maximum degree 5 are 6listedgecolorable. He proved that every planar graph with is of class 1 there are more general results, see 2 and 3 and then conjectured that every planar graph with maximum degree 6 or 7 is of class 1. Proof trivially, the theorem holds when n 3, so we may assume n. Pdf strong edgecoloring of planar graphs roman sotak and. List edge and list total colorings of planar graphs without non. Remove edges av, bv, and reduce any multiedges just created. The reductions are identi ed by means of a collection of con gurations, constant size subgraphs, one of which is always present in a planar graph. Listedge coloring planar graphs with bounded maximum degree.
In graph theory, an edge coloring of a graph is an assignment of colors to the edges of the graph so that no two incident edges have the same color. In this paper we prove that if g is a planar graph with maximum degree at most four. Coloring vertices and faces of locally planar graphs. Strong edgecolouring of sparse planar graphs request pdf. A graph gis uniquely edge kcolorable if there is an edge k coloring csuch that. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. An edge coloring of a graph is a function cfrom the edges of a graph to a set of colors having the property that if two edges share a common vertex as an endpoint, then cassigns them di erent colors. Planarity a graph is said to be planar if it can be drawn on a plane without any edges crossing. The strong chromatical index of a graph g is the least integer k such that g has a strongkedgecoloring, denoted by. As noted by vizing 1, if c 4, k 4, the octahedron, and the icosahedron have one edge subdivided each, class 2 planar graphs are produced for. Npcompleteness of list coloring and precoloring extension on. Edge coloring algorithms shinichi nakano, xiao zhou and takao nishizeki graduate school of information sciences tohoku university, sendal 98077, japan abstract. The edge coloring problem is one of the fundamental prob lems on graphs, which often appears in various scheduling problems like. A graph is 1planar if it can be drawn on the plane so that each edge is crossed by at most one other edge.
In addition, we prove that if g is a planar graph with maximum degree at least 4 and girth at least 7, then. Strong edge coloring for channel assignment in wireless radio networks. Strong edgecoloring was rst studied by fouquet and jolivet 1983, 1984 for cubic planar graphs. An edgekcoloring is an edgecoloring in which kcolors are used. Note if is a connected planar graph with edges and vertices, where, then. Both edge coloring and list edge coloring planar graphs are somewhat simpler. A structure of 1planar graph and its applications to. See 30 for an overview of algorithmic results on edge coloring. The graph g isplanarif it can be drawn in the plane without any edge crossing, where an edge crossing is an intersection point of two edges other than their common endpoints. Note that although typically an edge lies in the boundary of two faces, an edge can be in the boundary of a single face, which happens if and only if the edge is a cut edge. There are two monographs devoted to graph edge coloring. Such a drawing is called a planar representation of the graph. Hongfei fu sjtu jhc planar graphs and graph coloring nov. Our main result is a lineartime algorithm for coloring planar graphs with maximum degree.
On the equitable edgecoloring of 1planar graphs and planar. Pdf strong edge coloring of planar graphs researchgate. One form of the fourcolour theorem, due to tait 9, asserts that a 3regular planar graph can be 3edgecoloured if and only if it has no cutedge. By a simple application of eulers formula 16, it can be seen that graphs of bounded genus are o1inductive. G, is the minimum number of colors to construct such a coloring. Mathematics planar graphs and graph coloring geeksforgeeks. New lineartime algorithms for edgecoloring planar graphs. Currently most e cient algorithms for edgecoloring planar graphs.
On edge colorings of 1planar graphs without 5cycles with. An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. An s, trelaxed strong edge kcoloring is a mapping e g 1, 2, k such that for any edge e, there are at most s edges adjacent to e and t edges which are distance two apart from e assigned the same color as e. Acyclic edgecoloring of planar graphs siam journal on. Edgecolouring eightregular planar graphs maria chudnovsky1, katherine edwards2, paul seymour3 princeton university, princeton, nj 08544 january, 2012. All graphs used by the vertex coloring problem and con ictfree coloring problem are assumed to be simple graphs. An abstract graph that can be drawn as a plane graph is called a planar graph. In this thesis we prove that triangulations of maximum degree 5 are 6list edge colorable. Currently most e cient algorithms for edge coloring planar graphs. Planar graphs the drawing on the left is not a plane graph.
A graph is said to be planar if it can be drawn in a plane so that no edge cross. Now we return to the original graph coloring problem. Homomorphisms of 2edgecolored trianglefree planar graphs. A region is defined to be an area of the plane that is bounded by edges and cannot be further subdivided. An edgecoloring of a graph is a function cfrom the edges of a graph to a set of colors having the property that if two edges share a common vertex as an endpoint, then cassigns them di erent colors.
In this paper, it is proved that every 1planar graph with maximum degree. Especially, all the results can be applied to the list version of strong edge coloring. Edgecoloring algorithms shinichi nakano, xiao zhou and takao nishizeki graduate school of information sciences tohoku university, sendal 98077, japan abstract. Edgecolouring eightregular planar graphs princeton math. Later, the precise number of colors needed to color these graphs, in the worst case, was shown to be six. Edge colorings of planar graphs without 6cycles with two. A strong edge coloring of a graph is a proper edge coloring where the edges at distance at most two receive distinct colors. By eulers theorem, the number of regions which gives 12 regions. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a. In an analogous way, we can define the list version of strong chromatic index.
Other types of colorings on graphs also exist, most notably edge colorings that may be subject to various constraints. We show efficient algorithms for edgecoloring planar graphs. On edge colorings of 1planar graphs article pdf available in information processing letters 11. Guenin 5 introduced the notion of switching homomorphism for its. List strong edge coloring of planar graphs with maximum.
Thus the coloring is optimal for graphs with maximum degree 9. Strong edge coloring for channel assignment in wireless. A strongedgecoloring of a graph is a proper edge coloring in which every color class is an induced matching. Strong edgecoloring of subcubic planar graphs discrete. We also find necessary conditions for maximum degree to extend a list edge precoloring to eg for a planar graph g. That problem provided the original motivation for the. In the paper, we prove that every 1planar graph has an equitable edgecoloring with k colors for any integer. An edge k coloring is an edge coloring in which kcolors are used. We show that a planar graph with girth g and maximum degree. The terminology and notation used but undefined in this paper can be found. A strong edge coloring of a graph g is a proper edge coloring in which no two edges of the same color lie within distance 2 from each other.