Acyclic Chromatic Indices of Planar Graphs with Girth At Least 4

An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index of G is the smallest integer k such that G has an acyclic edge coloring using k colors. Fiamik (Math. Slovaca 28 (1978), 139–145) and later Alon et al. (J Graph Theory 37 (2001), 157–167) conjectured that for any simple graph G with maximum degree Δ. In this article, we confirm this conjecture for planar graphs of girth at least 4.     

平面图的绑定数

图G的绑定数b(G)是指边集合的最少边数,当这个边集合从G中去掉后所 得图的控制数大于G的控制数． Fischermann等人在[3]中给出了两个猜想: (1)如果 G是一个连通的平面图且围长g(G)≥4,则b(G)≤5;(2)如果G是一个连通的平面图且 围长g(G)≥5,则b(G)≤4．设n3表示度为3的顶点个数,r4和r5分别表示长为4和 5的圈的个数．本文,我们证明了如果r4<(5n3)/2 10,则猜想1成立;如果r5<12,则猜 想2成立．     

Maximal Induced Matchings in Triangle‐Free Graphs

An induced matching in a graph is a set of edges whose endpoints induce a 1‐regular subgraph. It is known that every n‐vertex graph has at most  maximal induced matchings, and this bound is the best possible. We prove that every n‐vertex triangle‐free graph has at most  maximal induced matchings; this bound is attained by every disjoint union of copies of the complete bipartite graph K_{3, 3}. Our result implies that all maximal induced matchings in an n‐vertex triangle‐free graph can be listed in time , yielding the fastest known algorithm for finding a maximum induced matching in a triangle‐free graph.     

Acyclic edge colorings of planar graphs and seriesparallel graphs

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G, denoted by a′(G), is the least number of colors in an acyclic edge coloring of G. Alon et al. conjectured that a′(G) ⩽ Δ(G) + 2 for any graphs. For planar graphs G with girth g(G), we prove that a′(G) ⩽ max{2Δ(G) − 2, Δ(G) + 22} if g(G) ⩾ 3, a′(G) ⩽ Δ(G) + 2 if g(G) ⩾ 5, a′(G) ⩽ Δ(G) + 1 if g(G) ⩾ 7, and a′(G) = Δ(G) if g(G) ⩾ 16 and Δ(G) ⩾ 3. For series-parallel graphs G, we have a′(G) ⩽ Δ(G) + 1. This work was supported by National Natural Science Foundation of China (Grant No. 10871119) and Natural Science Foundation of Shandong Province (Grant No. Y2008A20).     

系列平行图的围长和圆可选性

本文讨论了系列平行图的圆可选性．令x_(c,l)表示圆可选性(或圆列表着色数)．本文证明了围长至少是4n 1的系列平行图的圆可选性至多为2 1/n．     

大围长图的广义无圈染色

图的顶点染色称为是r-无圈的,如果它是正常染色,使得每一个圈C上顶点的颜色数至少为min{|C|,r}.图G的r-无圈染色数是图G的r-无圈染色中所用的最少的颜色数.我们证明了对于任意的r≥4,最大度为△、围长至少为2(r-1)△的图G的r-无圈染色数至多为6(r-1)△.     

不含相邻三角形的平面图的无圈边着色

An acyclic edge coloring of a graph G is a proper edge coloring such that there are no bichromatic cycles.The acyclic edge chromatic number of a graph G is the minimum number k such that there exists an acyclic edge coloring using k colors and is denoted by χ’ a(G).In this paper we prove that χ ’ a(G) ≤(G) + 5 for planar graphs G without adjacent triangles.     

关于平面图(3,1)~*-可选性的一个注记(英文)

给出了平面图的一个结构性定理,并证明了每个没有5-圈,相邻三角形,相邻四边形的平面图是(3,1)*-可选色的.     

On the Polarity and Monopolarity of Graphs

Polarity and monopolarity are properties of graphs defined in terms of the existence of certain vertex partitions; graphs with polarity and monopolarity are respectively called polar and monopolar graphs. These two properties commonly generalize bipartite and split graphs, but are hard to recognize in general. In this article we identify two classes of graphs, triangle‐free graphs and claw‐free graphs, restricting to which provide novel impact on the complexity of the recognition problems. More precisely, we prove that recognizing polarity or monopolarity remains NP‐complete for triangle‐free graphs. We also show that for claw‐free graphs the former is NP‐complete and the latter is polynomial time solvable. This is in sharp contrast to a recent result that both polarity and monopolarity can be recognized in linear time for line graphs. Our proofs for the NP‐completeness are simple reductions. The polynomial time algorithm for recognizing the monopolarity of claw‐free graphs uses a subroutine similar to the well‐known breadth‐first search algorithm and is based on a new structural characterization of monopolar claw‐free graphs, a generalization of one for monopolar line graphs obtained earlier.     

Large Induced Forests in Triangle-Free Planar Graphs

Given a planar graph G, what is the largest subset of vertices of G that induces a forest? Albertson and Berman [2] conjectured that every planar graph has an induced subgraph on at least half of the vertices that is a forest. For bipartite planar graphs, Akiyama and Wanatabe [1] conjectured that there is always an induced forest of size at least 5n/8. Here we prove that every triangle-free (and therefore every bipartite) planar graph on n vertices has an induced forest of size at least (17n+24)/32.     

Planar Digraphs without Large Acyclic Sets

Given a directed graph, an acyclic set is a set of vertices inducing a directed subgraph with no directed cycle. In this note, we show that for all integers , there exist oriented planar graphs of order n and digirth g for which the size of the maximum acyclic set is at most . When this result disproves a conjecture of Harutyunyan and shows that a question of Albertson is best possible.     

Independence,odd girth,and average degree

We prove several tight lower bounds in terms of the order and the average degree for the independence number of graphs that are connected and/or satisfy some odd girth condition. Our main result is the extension of a lower bound for the independence number of triangle‐free graphs of maximum degree at most three due to Heckman and Thomas [Discrete Math 233 (2001), 233–237] to arbitrary triangle‐free graphs. For connected triangle‐free graphs of order n and size m, our result implies the existence of an independent set of order at least (4n?m?1)/7. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:96‐111, 2011     

Induced Decompositions of Graphs

We consider those graphs G that admit decompositions into copies of a fixed graph F, each copy being an induced subgraph of G. We are interested in finding the extremal graphs with this property, that is, those graphs G on n vertices with the maximum possible number of edges. We discuss the cases where F is a complete equipartite graph, a cycle, a star, or a graph on at most four vertices.     

Gallai's Conjecture For Graphs of Girth at Least Four

In 1966, Gallai conjectured that for any simple, connected graph G having n vertices, there is a path‐decomposition of G having at most paths. In this article, we show that for any simple graph G having girth , there is a path‐decomposition of G having at most paths, where is the number of vertices of odd degree in G and is the number of nonisolated vertices of even degree in G.     

Cubic Graphs with Large Circumference Deficit

The circumference of a graph G is the length of a longest cycle. By exploiting our recent results on resistance of snarks, we construct infinite classes of cyclically 4‐, 5‐, and 6‐edge‐connected cubic graphs with circumference ratio bounded from above by 0.876, 0.960, and 0.990, respectively. In contrast, the dominating cycle conjecture implies that the circumference ratio of a cyclically 4‐edge‐connected cubic graph is at least 0.75. Up to our knowledge, no upper bounds on this ratio have been known before for cubic graphs with cyclic edge‐connectivity above 3. In addition, we construct snarks with large girth and large circumference deficit, solving Problem 1 proposed in [J. Hägglund and K. Markström, On stable cycles and cycle double covers of graphs with large circumference, Disc Math 312 (2012), 2540–2544].     

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$j\in\{6,7,8,9\}$.     

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.     

Generation of Cubic Graphs and Snarks with Large Girth

We describe two new algorithms for the generation of all non‐isomorphic cubic graphs with girth at least that are very efficient for and show how these algorithms can be restricted to generate snarks with girth at least k. Our implementation of these algorithms is more than 30, respectively 40 times faster than the previously fastest generator for cubic graphs with girth at least six and seven, respectively. Using these generators we have also generated all nonisomorphic snarks with girth at least six up to 38 vertices and show that there are no snarks with girth at least seven up to 42 vertices. We present and analyze the new list of snarks with girth 6.     

图的导出森林2-划分

对一个给定的简单图Ｇ，是否存在Ｖ（Ｇ）的一个２－划分（Ｖ１，Ｖ２）使得每个导出子图Ｇ［Ｖｉ］为森林？称该问题为导出森林２－划分问题．本文证明了对最大度为５的图该问题是ＮＰ－完全的，而对最大度≤４的图该问题多项式时间可解．