Subcubic Planar Graphs Research Articles

Packing and covering immersion-expansions of planar sub-cubic graphs

A graph H is an immersion of a graph G if H can be obtained by some subgraph G after lifting incident edges. We prove that there is a polynomial function f:N×N→N, such that if H is a connected planar sub-cubic graph on h>0 edges, G is a graph, and k is a non-negative integer, then either G contains k vertex/edge-disjoint subgraphs, each containing H as an immersion, or G contains a set F of f(k,h) vertices/edges such that G∖F does not contain H as an immersion.

Decycling with a matching

As a natural variant of the many decycling notions studied in graphs, we consider the problem to decide whether a given graph G has a matching M such that G−M is a forest. We establish NP-completeness of this problem for 2-connected planar subcubic graphs, and describe polynomial time algorithms that also determine such a matching if it exists for graphs that are claw- and paw-free, P5-free, chordal, and C4-free distance hereditary.

Strong edge-coloring of subcubic planar graphs

A strong[Formula: see text]-edge-coloring of a graph [Formula: see text] is a mapping [Formula: see text]: [Formula: see text], such that [Formula: see text] for every pair of distinct edges at distance at most two. The strong chromatical index of a graph [Formula: see text] is the least integer [Formula: see text] such that [Formula: see text] has a strong-[Formula: see text]-edge-coloring, denoted by [Formula: see text]. In this paper, we prove [Formula: see text] for any subcubic planar graph with [Formula: see text] and [Formula: see text]-cycles are not adjacent to [Formula: see text]-cycles.

On the algorithmic complexity of adjacent vertex closed distinguishing colorings number of graphs

An assignment of numbers to the vertices of graph G is closed distinguishing if for any two adjacent vertices v and u the sum of labels of the vertices in the closed neighborhood of the vertex v differs from the sum of labels of the vertices in the closed neighborhood of the vertex u unless they have the same closed neighborhood (i.e. N[u]=N[v]). The closed distinguishing number of a graph G, denoted by dis[G], is the smallest integer k such that there is a closed distinguishing labeling for G using integers from the set {1,2,…,k}. Also, for each vertex v∈V(G), let L(v) denote a list of natural numbers available at v. A list closed distinguishing labeling is a closed distinguishing labeling f such that f(v)∈L(v) for each v∈V(G). A graph G is said to be closed distinguishing k-choosable if every k-list assignment of natural numbers to the vertices of G permits a list closed distinguishing labeling of G. The closed distinguishing choice number of G, disℓ[G], is the minimum natural number k such that G is closed distinguishing k-choosable. In this work we show that for each integer t there is a bipartite graph G such that dis[G]>t. This is an answer to a question raised by Axenovich et al. in (Axenovich et al., 2016) that how ”dis” function depends on the chromatic number of a graph. It was shown that for every graph G with Δ≥2, dis[G]≤disℓ[G]≤Δ2−Δ+1 and also there are infinitely many values of Δ for which G might be chosen so that dis[G]=Δ2−Δ+1 (Axenovich et al., 2016). In this work, we prove that the difference between dis[G] and disℓ[G] can be arbitrary large and show that for every positive integer t there is a graph G such that disℓ[G]−dis[G]≥t. Also, we improve the current upper bound and give some number of upper bounds for the closed distinguishing choice number by using the Combinatorial Nullstellensatz. Among other results, we show that it is NP-complete to decide for a given planar subcubic graph G, whether dis[G]=2. Also, we prove that for every k≥3, it is NP-complete to decide whether dis[G]=k for a given graph G.

On the proper orientation number of bipartite graphs

An orientation of a graph G is a digraph D obtained from G by replacing each edge by exactly one of the two possible arcs with the same endvertices. For each v∈V(G), the indegree of v in D, denoted by dD−(v), is the number of arcs with head v in D. An orientation D of G is proper if dD−(u)≠dD−(v), for all uv∈E(G). The proper orientation number of a graph G, denoted by χ→(G), is the minimum of the maximum indegree over all its proper orientations. In this paper, we prove that χ→(G)≤(Δ(G)+Δ(G))/2+1 if G is a bipartite graph, and χ→(G)≤4 if G is a tree. It is well-known that χ→(G)≤Δ(G), for every graph G. However, we prove that deciding whether χ→(G)≤Δ(G)−1 is already an NP-complete problem on graphs with Δ(G)=k, for every k≥3. We also show that it is NP-complete to decide whether χ→(G)≤2, for planar subcubic graphs G. Moreover, we prove that it is NP-complete to decide whether χ→(G)≤3, for planar bipartite graphs G with maximum degree 5.

Classes of subcubic planar graphs for which the independent set problem is polynomially solvable

We prove the polynomial solvability of the independent set problem for some family of classes of the planar subcubic graphs.

On strong edge-colouring of subcubic graphs

A strong edge-colouring of a graph G is a proper edge-colouring such that every path of length 3 uses three different colours. In this paper we improve some previous results on the strong edge-colouring of subcubic graphs by showing that every subcubic graph with maximum average degree strictly less than 73 (resp. 52, 83, 207) can be strongly edge-coloured with six (resp. seven, eight, nine) colours. These upper bounds are optimal except the one of 83. Also, we prove that every subcubic planar graph without 4-cycles and 5-cycles can be strongly edge-coloured with nine colours.

Star list chromatic number of planar subcubic graphs

A proper coloring of the vertices of a graph G is called a star-coloring if the union of every two color classes induces a star forest. The graph G is L-star-colorable if for a given list assignment L there is a star-coloring ? such that ?(v)?L(v). If G is L-star-colorable for any list assignment L with |L(v)|?k for all v?V(G), then G is called k-star-choosable. The star list chromatic number of G, denoted by $\chi_{s}^{l}(G)$ , is the smallest integer k such that G is k-star-choosable. In this paper, we prove that every planar subcubic graph is 6-star-choosable.

Induced Matchings in Subcubic Planar Graphs

Induced Matchings in Subcubic Planar Graphs

2-distance 4-colorability of planar subcubic graphs with girth at least 22

2-distance 4-colorability of planar subcubic graphs with girth at least 22

Bounds and complexity results for strong edge colouring of subcubic graphs

Abstract A strong edge colouring of a graph G is a proper edge colouring such that every path of length 3 uses three colours. In this paper, we give some upper bounds for the minimum number of colours in a strong edge colouring of subcubic graphs as a function of the maximum average degree. We also prove the NP-completeness of the strong edge k-colouring problem for some restricted classes of subcubic planar graphs when k = 4 , 5 , 6 .

List 2-facial 5-colorability of plane graphs with girth at least 12

A proper vertex coloring of a plane graph is 2-facial if any two different vertices joined by a facial walk of length 2 are colored differently, and it is 2-distance if every two vertices at distance 2 from each other are colored differently. Note that any 2-facial coloring of a subcubic graph is 2-distance. It is known that every plane graph with girth at least 14 has a 2-facial 5-coloring [M. Montassier, A. Raspaud, A note on 2-facial coloring of plane graphs. Inform. Process. Lett. 98 (6) (2006) 235–241], and that every planar subcubic graph with girth at least 13 has a list 2-distance 5-coloring [F. Havet, Choosability of square of planar subcubic graphs with large girth, Discrete Math. 309 (2009) 3353–3563]. We strengthen these results by proving the list 2-facial 5-colorability of plane graphs with girth at least 12.

2-Distance 4-coloring of planar subcubic graphs

The trivial lower bound for the 2-distance chromatic number χ2(G) of any graph G with maximum degree Δ is Δ+1. It is known that χ2 = Δ+1 if the girth g of G is at least 7 and Δ is large enough. There are graphs with arbitrarily large Δ and g ≤ 6 having χ2(G) ≥ Δ+2. We prove the 2-distance 4-colorability of planar subcubic graphs with g ≥ 23, which strengthens a similar result by O. V. Borodin, A. O. Ivanova, and T. K. Neustroeva (2004) and Z. Dvořak, R. Skrekovski, and M. Tancer (2008) for g ≥ 24.

Injective colorings of planar graphs with few colors

An injective coloring of a graph is a vertex coloring where two vertices have distinct colors if a path of length two exists between them. In this paper some results on injective colorings of planar graphs with few colors are presented. We show that all planar graphs of girth ≥ 19 and maximum degree Δ are injectively Δ -colorable. We also show that all planar graphs of girth ≥ 10 are injectively ( Δ + 1 )-colorable, that Δ + 4 colors are sufficient for planar graphs of girth ≥ 5 if Δ is large enough, and that subcubic planar graphs of girth ≥ 7 are injectively 5-colorable.

Choosability of the square of planar subcubic graphs with large girth

We show that the choice number of the square of a subcubic graph with maximum average degree less than 18 / 7 is at most 6. As a corollary, we get that the choice number of the square of a subcubic planar graph with girth at least 9 is at most 6. We then show that the choice number of the square of a subcubic planar graph with girth at least 13 is at most 5.

List-Coloring Squares of Sparse Subcubic Graphs

The problem of coloring the square of a graph naturally arises in connection with the distance labelings, which have been studied intensively. We consider this problem for sparse subcubic graphs. We show that the choosability $\chi_\ell(G^2)$ of the square of a subcubic graph $G$ of maximum average degree $d$ is at most four if $d<24/11$ and $G$ does not contain a 5-cycle, at most five if $d<7/3$, and at most six if $d<5/2$. Wegner's conjecture claims that the chromatic number of the square of a subcubic planar graph is at most seven. Let $G$ be a planar subcubic graph of girth $g$. Our result implies that $\chi_\ell(G^2)$ is at most four if $g\ge 24$, at most $5$ if $g\ge 14$, and at most 6 if $g\ge 10$. For lower bounds, we find a planar subcubic graph $G_1$ of girth $9$ such that $\chi(G_1^2)=5$ and a planar subcubic graph $G_2$ of girth 5 such that $\chi(G_2^2)=6$. As a consequence, we show that the problem of 4-coloring of the square of a subcubic planar graph of girth $g=9$ is NP-complete. We conclude the paper by posing a few conjectures.

Linear choosability of graphs

A proper vertex coloring of a non-oriented graph G is linear if the graph induced by the vertices of any two color classes is a forest of paths. A graph G is linearly L-list colorable if for a given list assignment L = { L ( v ) : v ∈ V ( G ) } , there exists a linear coloring c of G such that c ( v ) ∈ L ( v ) for all v ∈ V ( G ) . If G is linearly L-list colorable for any list assignment with | L ( v ) | ⩾ k for all v ∈ V ( G ) , then G is said to be linearly k-choosable. In this paper, we investigate the linear choosability for some families of graphs: graphs with small maximum degree, with given maximum average degree, outerplanar and planar graphs. Moreover, we prove that deciding whether a bipartite subcubic planar graph is linearly 3-colorable is an NP-complete problem.

Two New Classes of Hamiltonian Graphs

Abstract We prove that a triangular grid without local cuts is (almost) always Hamiltonian. This suggests an efficient scheme for rendering triangulated manifolds by graphics hardware. We also show that the Hamiltonian Cycle problem is NP-Complete for planar subcubic graphs of arbitrarily high girth. As a by-product we prove that there exist tri-Hamiltonian planar subcubic graphs of arbitrarily high girth.

Packing [1, Δ]-factors in graphs of small degree

Given an undirected, connected graph G with maximum degree Δ, we introduce the concept of a [1, Δ]-factor k-packing in G, defined as a set of k edge-disjoint subgraphs of G such that every vertex of G has an incident edge in at least one subgraph. The problem of deciding whether a graph admits a [1,Δ]-factor k-packing is shown to be solvable in linear time for k = 2, but NP-complete for all k≥ 3. For k = 2, the optimisation problem of minimising the total number of edges of the subgraphs of the packing is NP-hard even when restricted to subcubic planar graphs, but can in general be approximated within a factor of \(\frac{42\Delta -30}{35\Delta-21}\) by reduction to the Maximum 2-Edge-Colorable Subgraph problem. Finally, we discuss implications of the obtained results for the problem of fault-tolerant guarding of a grid, which provides the main motivation for research.

Linear choosability of graphs

A proper vertex coloring of a non oriented graph $G=(V,E)$ is linear if the graph induced by the vertices of two color classes is a forest of paths. A graph $G$ is $L$-list colorable if for a given list assignment $L=\{L(v): v∈V\}$, there exists a proper coloring $c$ of $G$ such that $c(v)∈L(v)$ for all $v∈V$. If $G$ is $L$-list colorable for every list assignment with $|L(v)|≥k$ for all $v∈V$, then $G$ is said $k$-choosable. A graph is said to be lineary $k$-choosable if the coloring obtained is linear. In this paper, we investigate the linear choosability of graphs for some families of graphs: graphs with small maximum degree, with given maximum average degree, planar graphs... Moreover, we prove that determining whether a bipartite subcubic planar graph is lineary 3-colorable is an NP-complete problem.