Triangle-free Planar Graphs Research Articles

Edge choosability of planar graphs without small cycles

We investigate structural properties of planar graphs without triangles or without 4-cycles, and show that every triangle-free planar graph G is edge-( Δ ( G )+1)-choosable and that every planar graph with Δ ( G )≠5 and without 4-cycles is also edge-( Δ ( G )+1)-choosable.

Edge choosability of planar graphs without small cycles

We investigate structural properties of planar graphs without triangles or without 4-cycles, and show that every triangle-free planar graph G is edge-( Δ( G)+1)-choosable and that every planar graph with Δ( G)≠5 and without 4-cycles is also edge-( Δ( G)+1)-choosable.

Subcolorings and the subchromatic number of a graph

We consider the subchromatic number χ S( G) of graph G, which is the minimum order of all partitions of V( G) with the property that each class in the partition induces a disjoint union of cliques. Here we establish several bounds on subchromatic number. For example, we consider the maximum subchromatic number of all graphs of order n and in so doing answer a question posed in Jensen and Toft (Graph Coloring Problems, Wiley, New York, 1995). We also consider bounds on χ S( G) when the size and genus of G are known. We also consider the parameter when applied to planar and outerplanar graphs. It is known that the problem of determining whether χ S( G)⩽ k is NP-complete for all k⩾2. We extend this by showing it is NP-complete for k=2 even when restricted to the class of planar triangle-free graphs with maximum degree four. As a corollary, we see that showing a planar triangle-free graph of maximum degree four has a 1-defective chromatic number of two is NP-complete, answering a question of Cowen et al. (J. Graph Theory 24(3) (1997) 205–219). We show that determining whether χ S( G)⩽3 is NP-complete for planar graphs. We consider the subchromatic number of cartesian products of complete graphs and show a correspondence with a natural covering of matrices. We close by producing bounds on the subchromatic number in terms of chromatic number as well as the product of clique number with chromatic number. Sharpness for graphs with fixed clique size is discussed.

A short list color proof of Grötzsch's theorem

We give a short proof of the result that every planar graph of girth 5 is 3-choosable and hence also of Grötzsch's theorem saying that every planar triangle-free graph is 3-colorable.

Graph Subcolorings: Complexity and Algorithms

In a graph coloring, each color class induces a disjoint union of isolated vertices. A graph subcoloring generalizes this concept, since here each color class induces a disjoint union of complete graphs. Erdos and, independently, Albertson et al., proved that every graph of maximum degree at most 3 has a 2-subcoloring. We point out that this fact is best possible with respect to degree constraints by showing that the problem of recognizing 2-subcolorable graphs with maximum degree 4 is NP-complete, even when restricted to triangle-free planar graphs. Moreover, in general, for fixed k, recognizing k-subcolorable graphs is NP-complete on graphs with maximum degree at most k2 . In contrast, we show that, for arbitrary k, k-SUBCOLORABILITY can be decided in linear time on graphs with bounded treewidth and on graphs with bounded cliquewidth (including cographs as a specific case).

Triangle-Free Planar Graphs and Segment Intersection Graphs

Triangle-Free Planar Graphs and Segment Intersection Graphs

The NP-completeness of (1, r)-subcolorability of cubic graphs

A partition of the vertices of a graph G into k pairwise disjoint sets V 1,…, V k is called an ( r 1,…, r k )- subcoloring if the subgraph G i of G induced by V i, 1⩽i⩽k , consists of disjoint complete subgraphs, each of which has cardinality no more than r i . Due to Erdős and Albertson et al., independently, every cubic (i.e., 3-regular) graph has a (2,2)-subcoloring. Albertson et al. then asked for cubic graphs having (1,2)-subcolorings. We point out in this paper that this question is algorithmically difficult by showing that recognizing (1,2)-subcolorable cubic graphs is NP-complete, even when restricted to triangle-free planar graphs.

A Grötzsch-Type Theorem for List Colourings with Impropriety One

A graph G is m-choosable with impropriety d, or simply (m, d)*-choosable, if, for every list assignment L, where [mid ]L(v)[mid ][ges ]m for every v∈V(G), there exists an L-colouring of G such that each vertex of G has at most d neighbours coloured with the same colour as itself. We prove a Grötzsch-type theorem for list colourings with impropriety one, that is, the (3, 1)*-choosability for triangle-free planar graphs; in the proof the method of extending a precolouring of a 4- or 5-cycle is used.

The complexity of planar graph choosability

A graph G is k-choosable if for every assignment of a set S( v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S( v). We consider the complexity of deciding whether a given graph is k-choosable for some constant k. In particular, it is shown that deciding whether a given planar graph is 4-choosable is NP-hard, and so is the problem of deciding whether a given planar triangle-free graph is 3-choosable. We also obtain simple constructions of a planar graph which is not 4-choosable and a planar triangle-free graph which is not 3-choosable.

Star chromatic numbers of graphs

We investigate the relation between the star-chromatic number χ(G) and the chromatic number χ(G) of a graphG. First we give a sufficient condition for graphs under which their starchromatic numbers are equal to their ordinary chromatic numbers. As a corollary we show that for any two positive integersk, g, there exists ak-chromatic graph of girth at leastg whose star-chromatic number is alsok. The special case of this corollary withg=4 answers a question of Abbott and Zhou. We also present an infinite family of triangle-free planar graphs whose star-chromatic number equals their chromatic number. We then study the star-chromatic number of An infinite family of graphs is constructed to show that for each ε>0 and eachm≥2 there is anm-connected (m+1)-critical graph with star chromatic number at mostm+ε. This answers another question asked by Abbott and Zhou.

Star chromatic number of triangle-free planar graphs

The star chromatic number X ∗( G), introduced by Vince [1], of a graph G = ( V,E) is the least rational number k/d such that for V → [ k], the modular distance between the colors of any two adjacent vertices is at least d. We present an infinite family of triangle-free planar graphs and show that the star chromatic number of these graphs is 3.

Planar Ramsey Numbers

The planar Ramsey number PR( k, l) ( k, l ≥ 2) is the smallest integer n such that any planar graph on n vertices contains either a complete graph on k vertices or an independent set of size l. We find exact values of PR( k, l) for all k and l. Included is a proof of a 1976 conjecture due to Albertson, Bollobás, and Tucker that every triangle-free planar graph on n vertices contains an independent set of size ⌊ n/3⌋ + 1.

Every triangle-free planar graph has a planar upward drawing

A planar ordered set has a triangle-free, planar covering graph; on the other hand, there are nonplanar ordered sets whose covering graphs are planar. We show thatevery triangle-free planar graph has a planar upward drawing. This planar upward drawing can be constructed in time, polynomial in the number of vertices. Our results shed light on the apparently difficult problem, of long-standing, whether there is aneffective planarity-testing procedure for an ordered set.

On the net-embeddability of graphs

This paper provides two kinds of forbidden configurations for the rectilinear O-embeddability of triangle free planar graphs. Meanwhile, the characterizations of the O-embeddability for outerplanar graphs and Halin graphs are found.

The total interval number of a graph

A representation f of a graph G is a mapping f which assigns to each vertex of G a non-empty collection of intervals on the real line so that two distinct vertices x and y are adjacent if and only if there are intervals I ∈ f( x) and J ∈ f( y) with I ⌢ J ≠ ∅. We study the total interval number of G, defined as I( G) = min{Σ x ∈ V( G) | f( x)|: f is a representation of G}. Let n be the number of vertices of G. Our main results on I( G) are: (a) If G is a tree then I(G) ≤ ⌊ 5n 4 − 3 4 ⌋ for (c) If G is a triangle-free planar graph then I( G) ≤ 2 n − 3 for n ≥ 3. (d) If G is any triangle-free graph then I(G) ≤ ⌊ n 2 ⌋ ⌊ n 2 ⌋ + 1 . All bounds are sharp and for n ≥ 4 the bound of (d) is achieved iff G = K ⌊ n 2 ⌋,⌊ n 2 ⌋ . We conjecture that results (b)-(d) remain valid without the hypothesis “triangle-free” and present some partial results on this conjecture. We close with a result on the related concept of the depth-two total interval number and some open problems.