Triangle-free Planar Graphs Research Articles

Decomposing a triangle-free planar graph into a forest and a subcubic forest

We strengthen a result of Dross et al. (2017) that the vertex set of every triangle-free planar graph can be decomposed into a set that induces a forest and a set that induces a forest with maximum degree at most 5, showing that 5 can be replaced by 3.

Odd-sum colorings of planar graphs

A coloring of a graph G is a map f:V(G)→Z+ such that f(v)≠f(w) for all vw∈E(G). A coloring f is an odd-sum coloring if ∑w∈N[v]f(w) is odd, for each vertex v∈V(G). The odd-sum chromatic number of a graph G, denoted χos(G), is the minimum number of colors used (that is, the minimum size of the range) in an odd-sum coloring of G. Caro, Petruševski, and Škrekovski showed, among other results, that χos(G) is well-defined for every finite graph G and , in fact, χos(G)≤2χ(G). Thus, χos(G)≤8 for every planar graph G (by the 4 Color Theorem), χos(G)≤6 for every triangle-free planar graph G (by Grötzsch’s Theorem), and χos(G)≤4 for every bipartite graph.Caro et al. asked, for every even Δ≥4, whether there exists gΔ such that if G is planar with maximum degree Δ and girth at least gΔ then χos(G)≤5. They also asked, for every even Δ≥4, whether there exists gΔ such that if G is planar and bipartite with maximum degree Δ and girth at least gΔ then χos(G)≤3. We answer both questions negatively. We also refute a conjecture they made, resolve one further problem they posed, and make progress on another.

Weak diameter coloring of graphs on surfaces

Consider a graph G drawn on a fixed surface, and assign to each vertex a list of colors of size at least two if G is triangle-free and at least three otherwise. We prove that we can give each vertex a color from its list so that each monochromatic connected subgraph has bounded weak diameter (i.e., diameter measured in the metric of the whole graph G, not just the subgraph). In case that G has bounded maximum degree, this implies that each connected monochromatic subgraph has bounded size. This solves a problem of Esperet and Joret for planar triangle-free graphs, and extends known results in the general case to the list setting, answering a question of Wood.

Recoloring Planar Graphs of Girth at Least Five

For a positive integer , the -recoloring graph of a graph has as vertex set all proper -colorings of with two -colorings being adjacent if they differ by the color of exactly one vertex. A result of Dyer et al. regarding graphs of bounded degeneracy implies that the -recoloring graphs of planar graphs, the 5-recoloring graphs of triangle-free planar graphs and the 4-recoloring graphs planar graphs of girth at least six are connected. On the other hand, there are planar graphs whose 6-recoloring graph is disconnected, triangle-free planar graphs whose 4-recoloring graph is disconnected, and planar graphs of any given girth whose 3-recoloring graph is disconnected. The main result of this paper consists in showing, via a novel application of the discharging method, that the 4-recoloring graph of every planar graph of girth five is connected. This completes the classification of the connectedness of the recoloring graph for planar graphs of given girth. We also prove some theorems regarding the diameter of the recoloring graph of planar graphs.

Triangle-free planar graphs with at most [formula omitted] 3-colorings

Thomassen conjectured that triangle-free planar graphs have exponentially many 3-colorings. Recently, he disproved his conjecture by providing examples of such graphs with n vertices and at most 215n/log2⁡n 3-colorings. We improve his construction, giving examples of such graphs with at most 64nlog9/2⁡3<64n0.731 3-colorings. We conjecture this exponent is optimal.

Further extensions of the Grötzsch Theorem

The Grötzsch Theorem states that every triangle-free planar graph admits a proper 3-coloring. Among many of its generalizations, the one of Grünbaum and Aksenov, giving 3-colorability of planar graphs with at most three triangles, is perhaps the most known. A lot of attention was also given to extending 3-colorings of subgraphs to the whole graph. In this paper, we consider 3-colorings of planar graphs with at most one triangle. Particularly, we show that precoloring of any two non-adjacent vertices and precoloring of a face of length at most 4 can be extended to a 3-coloring of the graph. Additionally, we show that for every vertex of degree at most 3, a precoloring of its neighborhood with the same color extends to a 3-coloring of the graph. The latter result implies an affirmative answer to a conjecture on adynamic coloring. All the presented results are tight.

On the signed chromatic number of some classes of graphs

A signed graph (G,σ) is a graph G along with a function σ:E(G)→{+,−}. A closed walk of a signed graph is positive (resp., negative) if it has an even (resp., odd) number of negative edges, counting repetitions. A homomorphism of a (simple) signed graph to another signed graph is a vertex-mapping that preserves adjacencies and signs of closed walks. The signed chromatic number of a signed graph (G,σ) is the minimum number of vertices |V(H)| of a signed graph (H,π) to which (G,σ) admits a homomorphism.Homomorphisms of signed graphs have been attracting growing attention in the last decades, especially due to their strong connections to the theories of graph coloring and graph minors. These homomorphisms have been particularly studied through the scope of the signed chromatic number. In this work, we provide new results and bounds on the signed chromatic number of several families of signed graphs (planar graphs, triangle-free planar graphs, Kn-minor-free graphs, and bounded-degree graphs).

On the role of 3s for the 1-2-3 Conjecture

The 1-2-3 Conjecture states that every connected graph different from K2 admits a proper 3-(edge-)labelling, i.e., can have its edges labelled with 1,2,3 so that no two adjacent vertices are incident to the same sum of labels. In connection with some recent optimisation variants of this conjecture, in this paper we investigate the role of the label 3 in proper 3-labellings of graphs. An intuition from previous investigations is that, in general, it should always be possible to produce proper 3-labellings assigning label 3 to a only few edges.We prove that, for every p≥0, there are various graphs needing at least p 3s in their proper 3-labellings. Actually, deciding whether a given graph can be properly 3-labelled with p 3s is NP-complete for every p≥0. We also focus on classes of 3-chromatic graphs. For various classes of such graphs (cacti, cubic graphs, triangle-free planar graphs, etc.), we prove that there is no p≥1 such that all their graphs admit proper 3-labellings assigning label 3 to at most p edges. In such cases, we provide lower and upper bounds on the number of 3s needed.

Intersections and circuits in sets of line segments

Sets of straight line segments with special structures and properties appear in various applications of geometric modeling, such as scientific visualization, computer-aided design, and medical image processing. In this paper, we derive sharp upper and lower bounds on the number of intersection points and closed regions that can occur in sets of line segments with certain structure, in terms of the number of segments. In particular, we consider sets of segments whose underlying planar graphs are Halin graphs, cactus graphs, maximal planar graphs, and triangle-free planar graphs, as well as randomly produced segment sets.

Exponentially many 3-colorings of planar triangle-free graphs with no short separating cycles

The number of proper vertex-3-colorings of every triangle-free planar graph with n vertices and with no separating cycle of length 4 or 5 is at least 2n/17700000. On the other hand, for infinitely many n, there exists a triangle-free planar graph with separating cycles of length 4 and 5 whose number of proper vertex-3-colorings is <215n/log2⁡(n).

Injective coloring of planar graphs

An injective k-coloring of a graph G is a mapping f:V(G)→{1,2,…,k} such that for any two vertices v1,v2∈V(G), f(v1)≠f(v2) if N(v1)∩N(v2)≠∅. The injective chromatic number of a graph G, denoted by χi(G), is the smallest integer k such that G has an injective k-coloring. In this paper, we prove that for a Halin graph G, χi(G)≤Δ(G)+2. Moreover, χi(G)≤Δ(G)+1 if Δ(G)≥6. Also, we show that for a triangle-free planar graph G without intersecting 4-cycles, χi(G)≤Δ(G)+6 if Δ(G)≥20.

3-List-coloring graphs of girth at least five on surfaces

Grötzsch proved that every triangle-free planar graph is 3-colorable. Thomassen proved that every planar graph of girth at least five is 3-choosable. As for other surfaces, Thomassen proved that there are only finitely many 4-critical graphs of girth at least five embeddable in any fixed surface. This implies a linear-time algorithm for deciding 3-colorablity for graphs of girth at least five on any fixed surface. Dvořák, Král' and Thomas strengthened Thomassen's result by proving that the number of vertices in a 4-critical graph of girth at least five is linear in its genus. They used this result to prove Havel's conjecture that a planar graph whose triangles are pairwise far enough apart is 3-colorable. As for list-coloring, Dvořák proved that a planar graph whose cycles of size at most four are pairwise far enough part is 3-choosable.In this article, we generalize these results. First we prove a linear isoperimetric bound for 3-list-coloring graphs of girth at least five. Many new results then follow from the theory of hyperbolic families of graphs developed by Postle and Thomas. In particular, it follows that there are only finitely many 4-list-critical graphs of girth at least five on any fixed surface, and that in fact the number of vertices of a 4-list-critical graph is linear in its genus. This provides independent proofs of the above results while generalizing Dvořák's result to graphs on surfaces that have large edge-width and yields a similar result showing that a graph of girth at least five with crossings pairwise far apart is 3-choosable. Finally, we generalize to surfaces Thomassen's result that every planar graph of girth at least five has exponentially many distinct 3-list-colorings. Specifically, we show that every graph of girth at least five that has a 3-list-coloring has 2Ω(n)−O(g) distinct 3-list-colorings.

Choosability with union separation of triangle-free planar graphs

For a graph G and a positive integer k, a k-list assignment of G is a function L on the vertices of G such that for each vertex v∈V(G), |L(v)|≥k. Let s be a nonnegative integer. Then L is a (k,k+s)-list assignment of G if |L(u)∪L(v)|≥k+s for each edge uv. If for each (k,k+s)-list assignment L of G, G admits a proper coloring φ such that φ(v)∈L(v) for each v∈V(G), then we say G is (k,k+s)-choosable. This refinement of choosability is called choosability with union separation by Kumbhat et al. (2018), who showed that all planar graphs are (3,11)-choosable and (4,9)-choosable. In this paper, we prove that every triangle-free planar graph is (3,7)-choosable. We also prove that every planar graph with girth at least 5 is (2,7)-choosable.

Large independent sets in triangle-free cubic graphs: beyond planarity

The _independence ratio_ of a graph is the ratio of the size of its largest independent set to its number of vertices. Trivially, the independence ratio of a k-colorable graph is at least $1/k$ as each color class of a k-coloring is an independent set. However, better bounds can often be obtained for well-structured classes of graphs. In particular, Albertson, Bollobás and Tucker conjectured in 1976 that the independence ratio of every triangle-free subcubic planar graph is at least $3/8$. The conjecture was proven by Heckman and Thomas in 2006, and the ratio is best possible as there exists a cubic triangle-free planar graph with 24 vertices and the independence number equal to 9. The present article removes the planarity assumption. However, one needs to introduce an additional assumption since there are known to exist six 2-connected (non-planar) triangle-free subcubic graphs with the independence ratio less than $3/8$. Bajnok and Brinkmann conjectured that every 2-connected triangle-free subcubic graph has the independence ratio at least $3/8$ unless it is one of the six exceptional graphs. Fraughnaugh and Locke proposed a stronger conjecture: every triangle-free subcubic graph that does not contain one of the six exceptional graphs as a subgraph has independence ratio at least $3/8$. The authors prove these two conjectures, which implies in particular the result by Heckman and Thomas.

Flow extensions and group connectivity with applications

We study the flow extension of graphs, i.e., pre-assigning a partial flow on the edges incident to a given vertex and aiming to extend to the entire graph. This is closely related to Tutte’s 3-flow conjecture(1972) that every 4-edge-connected graph admits a nowhere-zero 3-flow and a Z3-group connectivity conjecture(3GCC) of Jaeger, Linial, Payan, and Tarsi(1992) that every 5-edge-connected graph G is Z3-connected. Our main results show that these conjectures are equivalent to their natural flow extension versions and present some applications. The 3-flow case gives an alternative proof of Kochol’s result(2001) that Tutte’s 3-flow conjecture is equivalent to its restriction on 5-edge-connected graphs and is implied by the 3GCC. It also shows a new fact that Grötzsch’s theorem (that triangle-free planar graphs are 3-colorable) is equivalent to its seemly weaker girth five case that planar graphs of girth 5 are 3-colorable. Our methods allow to verify 3GCC for graphs with crossing number one, which is in fact reduced to the planar case proved by Richter, Thomassen and Younger(2017). Other equivalent versions of 3GCC and related partial results are obtained as well.

Homomorphisms of partial t-trees and edge-colorings of partial 3-trees

A reformulation of the four color theorem is to say that K4 is the smallest graph to which every planar (loop-free) graph admits a homomorphism. Extending this theorem, the second author has proved (using the four color theorem) that the Clebsch graph (a well known triangle-free graph on 16 vertices) is a smallest graph to which every triangle-free planar graph admits a homomorphism. As a further generalization he has proposed that the projective cube of dimension 2k, PC(2k), (that is the Cayley graph (Z22k,{e1,e2,…,e2k,J}, where the ei's are the standard basis and J=e1+e2+⋯+e2k) is a smallest graph of odd-girth 2k+1 to which every planar graph of odd-girth at least 2k+1 admits a homomorphism. This conjecture is related to a conjecture of P. Seymour who claims that the fractional edge-chromatic number of a planar multigraph determines its edge-chromatic number (more precisely, Seymour conjectured that χ′(G)=⌈χf′(G)⌉ for any planar multigraph G). Note that the restriction of Seymour's conjecture to cubic (planar) graphs is Tait's reformulation of the four color theorem.Both these conjectures are believed to be true for the larger class of K5-minor-free graphs (which includes the class of planar graphs). Motivated by these conjectures and in extension of a recent work of L. Beaudou, F. Foucaud and the second author, which studies homomorphism bounds for the class of K4-minor-free graphs, in this work we first give a necessary and sufficient condition for a graph B of odd-girth 2k+1 to admit a homomorphism from any partial t-tree of odd-girth at least 2k+1. Applying our results to the class of partial 3-trees, which is a rich subclass of K5-minor-free graphs, we prove that PC(2k) is in fact a smallest graph of odd-girth 2k+1 to which every partial 3-tree of odd-girth at least 2k+1 admits a homomorphism. We then apply this result to show that every planar (2k+1)-regular multigraph G whose dual is a partial 3-tree, and whose fractional edge-chromatic number is 2k+1, is (2k+1)-edge-colorable. Both these results are the best known supports for the general cases of the above mentioned conjectures in extension of the four color theorem.

Complexity of Fall Coloring for Restricted Graph Classes

We strengthen a result by Laskar and Lyle (Discrete Appl. Math. (2009), 330-338) by proving that it is NP-complete to decide whether a bipartite planar graph can be partitioned into three independent dominating sets. In contrast, we show that this is always possible for every maximal outerplanar graph with at least three vertices. Moreover, we extend their previous result by proving that deciding whether a bipartite graph can be partitioned into $k$ independent dominating sets is NP-complete for every $k \geq 3$. We also strengthen a result by Henning et al. (Discrete Math. (2009), 6451-6458) by showing that it is NP-complete to determine if a graph has two disjoint independent dominating sets, even when the problem is restricted to triangle-free planar graphs. Finally, for every $k \geq 3$, we show that there is some constant $t$ depending only on $k$ such that deciding whether a $k$-regular graph can be partitioned into $t$ independent dominating sets is NP-complete. We conclude by deriving moderately exponential-time algorithms for the problem.

Neighbor Sum Distinguishing Total Coloring of Triangle Free IC-planar Graphs

A graph is IC-planar if it admits a drawing in the plane such that each edge is crossed at most once and two crossed edges share no common end-vertex. A proper total-k-coloring of G is called neighbor sum distinguishing if ∑c(u) ≠ ∑c(v) for each edge uv ∈ E(G), where ∑c(v) denote the sum of the color of a vertex v and the colors of edges incident with v. The least number k needed for such a total coloring of G, denoted by X′′∑(G), is the neighbor sum distinguishing total chromatic number. Pilśniak and Woźniak conjectured X′′∑(G) ≤ △(G) + 3 for any simple graph with maximum degree △(G). By using the famous Combinatorial Nullstellensatz, we prove that above conjecture holds for any triangle free IC-planar graph with △(G) ≥ 7. Moreover, it holds for any triangle free planar graph with △(G) ≥ 6.

Fractional Coloring of Planar Graphs of Girth Five

A graph $G$ is $(a:b)$-colorable if there exists an assignment of $b$-element subsets of $\{1,\ldots,a\}$ to vertices of $G$ such that sets assigned to adjacent vertices are disjoint. We first show that for every triangle-free planar graph $G$ and a vertex $x\in V(G)$, the graph $G$ has a set coloring $\varphi$ by subsets of $\{1,\ldots,6\}$ such that $|\varphi(v)|\geq 2$ for $v\in V(G)$ and $|\varphi(x)|=3$. As a corollary, every triangle-free planar graph on $n$ vertices is $(6n:2n+1)$-colorable. We further use this result to prove that for every $\Delta$, there exists a constant $M_{\Delta}$ such that every planar graph $G$ of girth at least five and maximum degree $\Delta$ is $(6M_{\Delta}:2M_{\Delta}+1)$-colorable. Consequently, planar graphs of girth at least five with bounded maximum degree $\Delta$ have fractional chromatic number at most $3-\frac{3}{2M_{\Delta}+1}$.

Exponentially Many Nowhere-Zero ℤ3-, ℤ4-, and ℤ6-Flows

We prove that, in several settings, a graph has exponentially many nowhere-zero flows. These results may be seen as a counting alternative to the well-known proofs of existence of ℤ3-, ℤ4-, and ℤ6-flows. In the dual setting, proving exponential number of 3-colorings of planar triangle-free graphs is a related open question due to Thomassen.