Decomposing a planar graph with girth at least 8 into a forest and a matching

W. He et al. showed that a planar graph of girth at least 11 can be decomposed into a forest and a matching. A. Bass et al. proved the same statement for planar graphs of girth at least 10. Recently, O.V. Borodin et al. improved the bound on the girth to 9. In this paper, we further improve the bound on the girth to 8. This bound is the best possible in the sense that there are planar graphs with girth 7 that cannot be decomposed into a forest and a matching.     

Partitioning a Planar Graph of Girth 10 into a Forest and a Matching

We prove that any finite planar graph with girth at least 10 can have its edges partitioned to form two graphs on the same vertices, one of which is a forest, and the other of which is a matching. Several related results are also demonstrated.     

Vertex decompositions of sparse graphs into an edgeless subgraph and a subgraph of maximum degree at most k

A graph G is (k,0)‐colorable if its vertices can be partitioned into subsets V₁ and V₂ such that in G[V₁] every vertex has degree at most k, while G[V₂] is edgeless. For every integer k?0, we prove that every graph with the maximum average degree smaller than (3k+4)/(k+2) is (k,0)‐colorable. In particular, it follows that every planar graph with girth at least 7 is (8, 0)‐colorable. On the other hand, we construct planar graphs with girth 6 that are not (k,0)‐colorable for arbitrarily large k. © 2009 Wiley Periodicals, Inc. J Graph Theory 65:83–93, 2010     

Partition of a planar graph with girth 6 into two forests with chain length at most 4

We prove that the vertex set of every planar graph with girth at most 6 can be partitioned into two subsets each of which generates a forest in which the length of each chain does not exceed 4.     

An Oriented 6-Coloring of Planar Graphs with Girth at Least 9

We prove that every oriented graph with a maximum average degree less than 18/7 admits a homomorphism into \(P_{7}^{*}\), the Paley tournament of order seven with one vertex deleted. In particular, every oriented planar graph of girth at least 9 has a homomorphism into \(P_{7}^{*}\), whence every planar graph of girth at least 9 has oriented chromatic number at most 6.     

Splitting Planar Graphs of Girth 6 into Two Linear Forests with Short Paths

Recently, Borodin, Kostochka, and Yancey (Discrete Math 313(22) (2013), 2638–2649) showed that the vertices of each planar graph of girth at least 7 can be 2‐colored so that each color class induces a subgraph of a matching. We prove that any planar graph of girth at least 6 admits a vertex coloring in colors such that each monochromatic component is a path of length at most 14. Moreover, we show a list version of this result. On the other hand, for each positive integer , we construct a planar graph of girth 4 such that in any coloring of vertices in colors there is a monochromatic path of length at least t. It remains open whether each planar graph of girth 5 admits a 2‐coloring with no long monochromatic paths.     

最大匹配图的围长

将一个图的所有最大匹配作为顶点集,称两个最大匹配相邻,若它们之一通过交换一条边得到另一个,由引所得图为该图的最大匹配图。本文研究了最大匹配图的围长,从而给出了最大匹配图是树或完全图的条件。     

Star coloring planar graphs from small lists

A star coloring of a graph is a proper vertex‐coloring such that no path on four vertices is 2‐colored. We prove that the vertices of every planar graph of girth 6 (respectively 7, 8) can be star colored from lists of size 8 (respectively 7, 6). We give an example of a planar graph of girth 5 that requires 6 colors to star color. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 324–337, 2010     

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.     

Decompositions of highly connected graphs into paths of length 3

We prove that a 171‐edge‐connected graph has an edge‐decomposition into paths of length 3 if and only its size is divisible by 3. It is a long‐standing problem whether 2‐edge‐connectedness is sufficient for planar triangle‐free graphs, and whether 3‐edge‐connectedness suffices for graphs in general. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 286–292, 2008     

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.     

无8-,9-和10-圈的平面图的3-可选择性

寻找平面图是3-或者4-可选择的充分条件是图的染色理论中一个重要研究课题,本文研究了围长至少是4的特殊平面图的选择数,通过权转移的方法证明了每个围长至少是4且不合8-圈,9-圈和10-圈的平面图是3-可选择的．     

最大度至多为6的平面图的L(2,1)-标号

令Δ(G),g(G)和λ(G)分别为图G的最大度,围长,和L(2,1)-标号数.证明了若G是Δ(G)≤6和g(G)≥5的平面图,则λ(G)≤Δ(G)+13.进而关于Δ(G)≤6和g(G)≥5的平面图G,这个界要比先前的结果好.     

Induced forests in cubic graphs

Bounds are obtained for the number of vertices in a largest induced forest in a cubic graph with large girth. In particular, as girth increases without bound, the ratio of the number of vertices in a largest induced forest to the number of vertices in the whole graph approaches $\text{3}\text{4}$.     

划分围长至少为6 的平面图为有界大小的分支

图$G$的$(\mathcal{O}_{k_1}, \mathcal{O}_{k_2})$-划分是将$V(G)$划分成两个非空子集$V_{1}$和$V_{2}$, 使得$G[V_{1}]$和$G[V_{2}]$分别是分支的阶数至多$k_1$和$k_2$的图.在本文中,我们考虑了有围长限制的平面图的点集划分问题,使得每个部分导出一个具有有界大小分支的图.我们证明了每一个围长至少为6并且$i$-圈不与$j$-圈相交的平面图允许$(\mathcal{O}_{2}$, $\mathcal{O}_{3})$-划分,其中$i\in\{6,7,8\}$和$j\in\{6,7,8,9\}$.     

Homomorphism Bounds for Oriented Planar Graphs of Given Minimum Girth

We find necessary conditions for a digraph H to admit a homomorphism from every oriented planar graph of girth at least n, and use these to prove the existence of a planar graph of girth 6 and oriented chromatic number at least 7. We identify a ${\overleftrightarrow{K_4}}$ -free digraph of order 7 which admits a homomorphism from every oriented planar graph (here ${\overleftrightarrow{K_n}}$ means a digraph with n vertices and arcs in both directions between every distinct pair), and a ${\overleftrightarrow{K_3}}$ -free digraph of order 4 which admits a homomorphism from every oriented planar graph of girth at least 5.     

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.     

List 2-distance (Δ + 1)-coloring of planar graphs with given girth

Some sufficient conditions (in terms of the girth and maximum degree) are given for the list 2-distance chromatic number of a planar graph with maximum degree Δ to be equal to Δ + 1.     

Partitions of Graphs into Cographs

Cographs from the minimal family of graphs containing K₁ which are closed with respect to complements and unions. We discuss vertex partitions of graphs into the smallest number of cographs, where the partition is as small as possible. We shall call the order of such a partition the c-chromatic number of the graph. We begin by axiomatizing several well-known graphical parameters as motivation for this function. We present several bounds on c-chromatic number in terms of well-known expressions. We show that if a graph is triangle-free, then its chromatic number is bounded between the c-chromatic number and twice this number. We show both bounds are sharp, for graphs with arbitrarily high girth. This provides an alternative proof to a result in [3]; there exist triangle-free graphs with arbitrarily large c-chromatic numbers. We show that any planar graph with girth at least 11 has a c-chromatic number of at most two. We close with several remarks on computational complexity. In particular, we show that computing the c-chromatic number is NP-complete for planar graphs.