Triangle-free Planar Graphs Research Articles

Storage Capacity as an Information-Theoretic Vertex Cover and the Index Coding Rate

Motivated by applications in distributed storage, the storage capacity of a graph was recently defined to be the maximum amount of information that can be stored across the vertices of a graph such that the information at any vertex can be recovered from the information stored at the neighboring vertices. Computing the storage capacity is a fundamental problem in network coding and is related, or equivalent, to some well-studied problems such as index coding with side information and generalized guessing games. In this paper, we consider storage capacity as a natural information-theoretic analogue of the minimum vertex cover of a graph. Indeed, while it was known that storage capacity is upper bounded by minimum vertex cover, we show that by treating it as such we can get a 3/2 approximation for planar graphs, and a 4/3 approximation for triangle-free planar graphs. Since the storage capacity is intimately related to the index coding rate, we get a 2 approximation of index coding rate for planar graphs and 3/2 approximation for triangle-free planar graphs. Previously, only a trivial 4 approximation of the index coding rate was known for planar graphs. We also show a polynomial time approximation scheme for the index coding rate when the alphabet size is constant. We then develop a general method of “gadget covering” to upper bound the storage capacity in terms of the average of a set of vertex covers. This method is intuitive and leads to the exact characterization of storage capacity for various families of graphs. As an illustrative example, we use this approach to derive the exact storage capacity of cycles-with-chords, a family of graphs related to outerplanar graphs. Finally, we generalize the storage capacity notion to include recovery from partial node failures in distributed storage. We show tight upper and lower bounds on this partial recovery capacity that scales nicely with the fraction of failures in a vertex.

Note on 3-choosability of planar graphs with maximum degree 4

Deciding whether a planar graph (even of maximum degree 4) is 3-colorable is NP-complete. Determining subclasses of planar graphs being 3-colorable has a long history, but since Grötzsch’s result that triangle-free planar graphs are such, most of the effort was focused to solving Havel’s and Steinberg’s conjectures. In this paper, we prove that every planar graph obtained as a subgraph of the medial graph of any bipartite plane graph is 3-choosable. These graphs are allowed to have close triangles (even incident), and have no short cycles forbidden, hence representing an entirely different class than the graphs inferred by the above mentioned conjectures.

Gallai’s path decomposition conjecture for triangle-free planar graphs

A path decomposition of a graph G is a collection of edge-disjoint paths of G that covers the edge set of G. Gallai (1968) conjectured that every connected graph on n vertices admits a path decomposition of cardinality at most ⌊(n+1)∕2⌋. Gallai’s Conjecture has been verified for many classes of graphs. In particular, Lovász (1968) verified this conjecture for graphs with at most one vertex with even degree, and Pyber (1996) verified it for graphs in which every cycle contains a vertex with odd degree. Recently, Bonamy and Perrett (2016) verified Gallai’s Conjecture for graphs with maximum degree at most 5, and Botler et al. (2017) verified it for graphs with treewidth at most 3. In this paper, we verify Gallai’s Conjecture for triangle-free planar graphs.

A lower bound on the order of the largest induced linear forest in triangle-free planar graphs

We prove that every triangle-free planar graph of order n and size m has an induced linear forest with at least 9n−2m11 vertices, and thus at least 5n+811 vertices. Furthermore, we show that there are triangle-free planar graphs on n vertices whose largest induced linear forest has order ⌈n2⌉+1.

Multiple list colouring triangle free planar graphs

This paper proves that for each positive integer m, there is a triangle free planar graph G which is not (3m+⌈m17⌉−1,m)-choosable.

Colouring of generalized signed triangle-free planar graphs

We view an undirected graph G as a symmetric digraph, where each edge xy is replaced by two opposite arcs e=(x,y) and e−1=(y,x). Assume S is an inverse closed subset of permutations of positive integers. We say G is S-k-colourable if for any mapping σ:E(G)→S with σ(x,y)=(σ(y,x))−1, there is a mapping f:V(G)→[k]={1,2,…,k} such that σe(f(x))≠f(y) for each arc e=(x,y). The concept of S-k-colourable is a common generalization of several other colouring concepts. This paper is focused on finding the sets S such that every triangle-free planar graph is S-3-colourable. Such a set S is called TFP-good. Grötzsch’s theorem is equivalent to say that S={id} is TFP-good. We prove that for any inverse closed subset S of S3 which is not isomorphic to {id,(12)}, S is TFP-good if and only if either S={id} or there exists a∈[3] such that for each π∈S, π(a)≠a. It remains an open question to determine whether or not S={id,(12)} is TFP-good.

Triangle-free planar graphs with small independence number

Since planar triangle-free graphs are 3-colourable, such a graph with n vertices has an independent set of size at least n∕3. We prove that unless the graph contains a certain obstruction, its independence number is at least n∕(3−ε) for some fixed ε>0. We also provide a reduction rule for this obstruction, which enables us to transform any plane triangle-free graph G into a plane triangle-free graph G′ such that α(G′)−|G′|∕3=α(G)−|G|∕3 and|G′|≤(α(G)−|G|∕3)∕ε. We derive a number of algorithmic consequences as well as a structural description of n-vertex plane triangle-free graphs whose independence number is close to n∕3.

Large induced forests in planar graphs with girth 4

We give a new lower bound on the order of a largest induced forest in planar graphs with girth 4. We prove that a triangle-free planar graph of order n admits an induced forest of order at least 6n+711, improving the lower bound of Salavatipour (2006).

A Better Bound on the Largest Induced Forests in Triangle-Free Planar Graph

It is well-known that there exists a triangle-free planar graph of n vertices that has the largest induced forest of order at most $$\frac{5n}{8}$$ . Salavatipour (Graphs Comb 22(1):113–126, 2006) proved that there is a forest of order at least $$\frac{5n}{9.41}$$ in any triangle-free planar graph of n vertices. Dross et al. (Large induced forests in planar graphs with girth 4 or 5, arXiv:1409.1348 , 2014) improved Salavatipour’s bound to $$\frac{5n}{9.17}$$ . In this work, we further improve the bound to $$\frac{5n}{9}$$ . Our technique is inspired by the recent ideas from Lukot’ka et al. (Electron J Comb 22(1):P1–P11, 2015).

A study on oriented relative clique number

An oriented graph is a directed graph with no directed cycle of length one or two. The relative clique number of an oriented graph is the cardinality of a largest subset X of vertices such that each pair of vertices is either adjacent or connected by a directed 2-path. It is known that the oriented relative clique number of a planar graph is at most 80. Here we improve the upper bound to 32. We also prove an upper bound of 14 for oriented relative clique number of triangle-free planar graphs. Furthermore, we determine the exact values of oriented relative clique number for the families of outerplanar graphs with girth at least g and planar graphs with girth at least g+2 for all g≥3. Moreover, we study the relation of oriented relative clique number with oriented chromatic number, oriented absolute clique number and maximum degree of a graph. We also show that oriented relative clique number of a connected subcubic graph is at most seven which weakly supports a conjecture by Sopena (JGT 1997).

Relative clique number of planar signed graphs

A signed relative clique number of signed graph (where edges are assigned positive or negative signs) is the size of a largest subset X of vertices such that every two vertices are either adjacent or are part of a 4-cycle with an odd number of negative edges. The signed relative clique number is sandwiched between two other parameters of signed graphs, namely, the signed absolute clique number and the signed chromatic number, all three notions defined in Naserasr et al. (2014). Thus, together with a result from Ochem et al. (2016) the lower bound of 8 and upper bound of 40 has already been proved for the signed relative clique number of the family of planar graphs. Here we improve the upper bound to 15. Furthermore, we determine the exact values of signed relative clique number of the families of outerplanar graphs and triangle-free planar graphs.

Induced 2-Degenerate Subgraphs of Triangle-Free Planar Graphs

A graph is $k$-degenerate if every subgraph has minimum degree at most $k$. We provide lower bounds on the size of a maximum induced 2-degenerate subgraph in a triangle-free planar graph. We denote the size of a maximum induced 2-degenerate subgraph of a graph $G$ by $\alpha_2(G)$. We prove that if $G$ is a connected triangle-free planar graph with $n$ vertices and $m$ edges, then $\alpha_2(G) \geq \frac{6n - m - 1}{5}$. By Euler's Formula, this implies $\alpha_2(G) \geq \frac{4}{5}n$. We also prove that if $G$ is a triangle-free planar graph on $n$ vertices with at most $n_3$ vertices of degree at most three, then $\alpha_2(G) \geq \frac{7}{8}n - 18 n_3$.

Fine Structure of 4-Critical Triangle-Free Graphs I. Planar Graphs with Two Triangles and 3-Colorability of Chains

Aksenov proved that in a planar graph $G$ with at most one triangle, every precoloring of a 4-cycle can be extended to a 3-coloring of $G$. We give an exact characterization of planar graphs with two triangles in which some precoloring of a 4-cycle does not extend. We apply this characterization to solve the precoloring extension problem from two 4-cycles in a triangle-free planar graph in the case that the precolored 4-cycles are separated by many disjoint 4-cycles. The latter result is used in follow-up papers [SIAM J. Discrete Math., 31 (2017), pp. 865--874; SIAM J. Discrete Math., 32 (2018), pp. 94--105] to give detailed information about the structure of 4-critical triangle-free graphs embedded in a fixed surface.

Do Triangle-Free Planar Graphs have Exponentially Many $3$-Colorings?

Thomassen conjectured that triangle-free planar graphs have an exponential number of $3$-colorings. We show this conjecture to be equivalent to the following statement: there exists a positive real $\alpha$ such that whenever $G$ is a planar graph and $A$ is a subset of its edges whose deletion makes $G$ triangle-free, there exists a subset $A'$ of $A$ of size at least $\alpha|A|$ such that $G-(A\setminus A')$ is $3$-colorable. This equivalence allows us to study restricted situations, where we can prove the statement to be true.

2-subcoloring is NP-complete for planar comparability graphs

A k-subcoloring of a graph is a partition of the vertex set into at most k cluster graphs, that is, graphs with no induced P3. 2-subcoloring is known to be NP-complete for comparability graphs and three subclasses of planar graphs, namely triangle-free planar graphs with maximum degree 4, planar perfect graphs with maximum degree 4, and planar graphs with girth 5. We show that 2-subcoloring is also NP-complete for planar comparability graphs with maximum degree 4.

3-coloring triangle-free planar graphs with a precolored 9-cycle

Given a triangle-free planar graph G and a 9-cycle C in G, we characterize situations where a 3-coloring of C does not extend to a proper 3-coloring of G. This extends previous results when C is a cycle of length at most 8.

Exponentially many nowhere-zero [formula omitted]-, [formula omitted]-, and [formula omitted]-flows

We prove that, in several settings, a graph has exponentially many nowhere-zero flows. Our 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.

Partitioning a triangle-free planar graph into a forest and a forest of bounded degree

An (F,Fd)-partition of a graph is a vertex-partition into two sets F and Fd such that the graph induced by F is a forest and the one induced by Fd is a forest with maximum degree at most d. We prove that every triangle-free planar graph admits an (F,F5)-partition. Moreover we show that if for some integer d there exists a triangle-free planar graph that does not admit an (F,Fd)-partition, then it is an NP-complete problem to decide whether a triangle-free planar graph admits such a partition.

Backbone coloring for triangle-free planar graphs

Let G be a graph and H a subgraph of G. A backbone-k-coloring of (G, H) is a mapping f: V(G) → {1, 2, ···, k} such that |f(u) − f(v)| ≥ 2 if uv ∈ E(H) and |f(u) − f(v)| ≥ 1 if uv ∈ E(G)E(H). The backbone chromatic number of (G, H) denoted by χ b (G, H) is the smallest integer k such that (G, H) has a backbone-k-coloring. In this paper, we prove that if G is either a connected triangle-free planar graph or a connected graph with mad(G) < 3, then there exists a spanning tree T of G such that χ b (G, T) ≤ 4.

Incidence coloring of graphs with high maximum average degree

An incidence of an undirected graph G is a pair (v,e) where v is a vertex of G and e an edge of G incident with v. Two incidences (v,e) and (w,f) are adjacent if one of the following holds: (i) v=w, (ii) e=f or (iii) vw=e or f. An incidence coloring of G assigns a color to each incidence of G in such a way that adjacent incidences get distinct colors. In 2005, Hosseini Dolama and Sopena (2005) proved that every graph with maximum average degree strictly less than 3 can be incidence colored with Δ+3 colors. Recently, Bonamy et al. (2014) proved that every graph with maximum degree at least 4 and with maximum average degree strictly less than 73 admits an incidence (Δ+1)-coloring. In this paper we give bounds for the number of colors needed to color graphs having maximum average degrees bounded by different values between 4 and 6. In particular we prove that every graph with maximum degree at least 7 and with maximum average degree less than 4 admits an incidence (Δ+3)-coloring. This result implies that every triangle-free planar graph with maximum degree at least 7 is incidence (Δ+3)-colorable. We also prove that every graph with maximum average degree less than 6 admits an incidence (Δ+7)-coloring. More generally, we prove that Δ+k−1 colors are enough when the maximum average degree is less than k and the maximum degree is sufficiently large.