Almost resolvable k‐cycle systems with

In this paper, we show that if and , then there exists an almost resolvable k‐cycle system of order for all except possibly for and . Thus we give a partial solution to an open problem posed by Lindner, Meszka, and Rosa (J. Combin. Des., vol. 17, pp. 404–410, 2009).     

On the existence of cycle frames and almost resolvable cycle systems

Suppose H is a complete m-partite graph K_m(n₁,n₂,…,n_m) with vertex set V and m independent sets G₁,G₂,…,G_m of n₁,n₂,…,n_m vertices respectively. Let G={G₁,G₂,…,G_m}. If the edges of λH can be partitioned into a set C of k-cycles, then (V,G,C) is called a k-cycle group divisible design with index λ, denoted by (k,λ)-CGDD. A (k,λ)-cycle frame is a (k,λ)-CGDD (V,G,C) in which C can be partitioned into holey 2-factors, each holey 2-factor being a partition of V?G_i for some G_i∈G. Stinson et al. have resolved the existence of (3,λ)-cycle frames of type g^u. In this paper, we show that there exists a (k,λ)-cycle frame of type g^u for k∈{4,5,6} if and only if , , u≥3 when k∈{4,6}, u≥4 when k=5, and (k,λ,g,u)≠(6,1,6,3). A k-cycle system of order n whose cycle set can be partitioned into (n−1)/2 almost parallel classes and a half-parallel class is called an almost resolvable k-cycle system, denoted by k-ARCS(n). Lindner et al. have considered the general existence problem of k-ARCS(n) from the commutative quasigroup for . In this paper, we give a recursive construction by using cycle frames which can also be applied to construct k-ARCS(n)s when . We also update the known results and prove that for k∈{3,4,5,6,7,8,9,10,14} there exists a k-ARCS(2kt+1) for each positive integer t with three known exceptions and four additional possible exceptions.     

Almost resolvable Pk-decompositions of complete graphs

An almost P_k-factor of G is a P_k-factor of G - { v } for some vertex v. An almost resolvable P_k-decomposition of λK_n is a partition of the edges of λK_n into almost P_k-factors. We prove that necessary and sufficient conditions for the existence of an almost resolvable P_k-decomposition of λK_n are n ≡ 1 (mod k) and λnk/2 ≡ 0 (mod k ?1).     

Completing the spectrum of almost resolvable cycle systems with odd cycle length

In this paper, we construct almost resolvable cycle systems of order $4k+1$ for odd $k\ge 11$. This completes the proof of the existence of almost resolvable cycle systems with odd cycle length. As a by-product, some new solutions to the Hamilton–Waterloo problem are also obtained.     

Acyclic 5-choosability of planar graphs with neither 4-cycles nor chordal 6-cycles

A proper vertex coloring of a graph G=(V,E) is acyclic if G contains no bicolored cycle. A graph G is acyclically L-list colorable if for a given list assignment L={L(v):v∈V}, there exists a proper acyclic coloring ? of G such that ?(v)∈L(v) for all v∈V(G). If G is acyclically L-list colorable for any list assignment with |L(v)|≥k for all v∈V, then G is acyclically k-choosable. In this paper it is proved that every planar graph with neither 4-cycles nor chordal 6-cycles is acyclically 5-choosable. This generalizes the results of [M. Montassier, A. Raspaud, W. Wang, Acyclic 5-choosability of planar graphs without small cycles, J. Graph Theory 54 (2007) 245-260], and a corollary of [M. Montassier, P. Ochem, A. Raspaud, On the acyclic choosability of graphs, J. Graph Theory 51 (4) (2006) 281-300].     

Planar graphs without 5-cycles or without 6-cycles

Let G be a planar graph without 5-cycles or without 6-cycles. In this paper, we prove that if G is connected and δ(G)≥2, then there exists an edge xy∈E(G) such that d(x)+d(y)≤9, or there is a 2-alternating cycle. By using the above result, we obtain that (1) its linear 2-arboricity , (2) its list total chromatic number is Δ(G)+1 if Δ(G)≥8, and (3) its list edge chromatic number is Δ(G) if Δ(G)≥8.     

Defective 2-colorings of planar graphs without 4-cycles and 5-cycles

A 2-coloring is a coloring of vertices of a graph with colors 1 and 2. Define $V_i?\{v\in V\left(G\right):c\left(v\right)=i\}$ for $i=1$ and $2.$ We say that $G$ is $(d_1,d_2)$-colorable if $G$ has a 2-coloring such that $V_i$ is an empty set or the induced subgraph $G\left[V_i\right]$ has the maximum degree at most $d_i$ for $i=1$ and $2.$ Let $G$ be a planar graph without 4-cycles and 5-cycles. We show that the problem to determine whether $G$ is $(0,k)$-colorable is NP-complete for every positive integer $k.$ Moreover, we construct non-$(1,k)$-colorable planar graphs without 4-cycles and 5-cycles for every positive integer $k.$ In contrast, we prove that $G$ is $(d_1,d_2)$-colorable where $(d_1,d_2)=(4,4),(3,5),$ and $(2,9).$     

最大度为7 且不含带弦5- 圈的平面图是8- 全可染的

若能用k种颜色给图的顶点和边同时进行染色使得相邻或相关联的元素(顶点或边) 染不同的色, 则称这个图是k- 全可染的. 显然, 给最大度为Δ的图进行全染色, 至少要用Δ + 1 种不同的色.本文证明最大度为7 且不含带弦5- 圈的平面图是8- 全可染的. 这一结果进一步拓广了(Δ+1)- 全可染图类.     

Packing cycles in complete graphs

We introduce a new technique for packing pairwise edge-disjoint cycles of specified lengths in complete graphs and use it to prove several results. Firstly, we prove the existence of dense packings of the complete graph with pairwise edge-disjoint cycles of arbitrary specified lengths. We then use this result to prove the existence of decompositions of the complete graph of odd order into pairwise edge-disjoint cycles for a large family of lists of specified cycle lengths. Finally, we construct new maximum packings of the complete graph with pairwise edge-disjoint cycles of uniform length.     

Edge choosability of planar graphs without 5-cycles with a chord

Let G be a plane graph having no 5-cycles with a chord. If either Δ≥6, or Δ=5 and G contains no 4-cycles with a chord or no 6-cycles with a chord, then G is edge-(Δ+1)-choosable, where Δ denotes the maximum degree of G.     

Acyclic 5-choosability of planar graphs without 4-cycles

A proper vertex coloring of a graph G=(V,E) is acyclic if G contains no bicolored cycle. A graph G is acyclically L-list colorable if for a given list assignment L={L(v):v∈V}, there exists a proper acyclic coloring π of G such that π(v)∈L(v) for all v∈V. If G is acyclically L-list colorable for any list assignment with |L(v)|≥k for all v∈V, then G is acyclically k-choosable. In this paper we prove that every planar graph without 4-cycles and without two 3-cycles at distance less than 3 is acyclically 5-choosable. This improves a result in [M. Montassier, P. Ochem, A. Raspaud, On the acyclic choosability of graphs, J. Graph Theory 51 (2006) 281-300], which says that planar graphs of girth at least 5 are acyclically 5-choosable.     

Almost regular edge colorings and regular decompositions of complete graphs

For k = 1 and k = 2, we prove that the obvious necessary numerical conditions for packing t pairwise edge‐disjoint k‐regular subgraphs of specified orders m₁,m₂,… ,m_t in the complete graph of order n are also sufficient. To do so, we present an edge‐coloring technique which also yields new proofs of various known results on graph factorizations. For example, a new construction for Hamilton cycle decompositions of complete graphs is given. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 499–506, 2008     

Packing complete multipartite graphs with 4‐cycles

In this paper we completely solve the problem of finding a maximum packing of any complete multipartite graph with edge‐disjoint 4‐cycles, and the minimum leaves are explicitly given. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 107–127, 2001     

Planar graphs with maximum degree 7 and without 5-cycles are 8-totally-colorable

It is proved that planar graphs with maximum degree 7 and without 5-cycles are 8-totally-colorable.     

不含5-圈和相邻4-圈的平面图的线性2-荫度的一个上界

图G的一个边分解是指将G分解成子图G_1,G_2,…,G_m使得E(G)=E(G_1)=∪E(G_2)∪…∪E(G_m),且对于i≠j,E(G_i)∩E(G_j)=?.一个线性k-森林是指每个分支都是长度最多为k的路的图.图G的线性k-荫度la_k(G)是使得G可以边分解为m个线性k-森林的最小整数m.显然,la_1(G)是G的边色数χ'(G); la_∞(G)表示每条分支路是无限长度时的情况,即通常所说的G的线性荫度la(G).利用权转移的方法研究平面图的线性2-荫度la_2(G).设G是不含有5-圈和相邻4-圈的平面图,证明了若G连通且δ(G)≥2,则G包含一条边xy使得d(x)+d(y)≤8或包含一个2-交错圈.根据这一结果得到其线性2-荫度的上界为[△/2]+4.     

Support sizes of threefold resolvable triple systems

Let SS_R(v, 3) denote the set of all integer b* such that there exists a RTS(v, 3) with b* distinct triples. In this paper, we determine the set SS_R(v, 3) for v ≡ 3 (mod 6) and v ≥ 3 with only five undecided cases. We establish that SS_R(v, 3) = P(v, 3) for v ≡ 3 (mod 6), v ≥ 21 and v ≠ 33, 39 where P(v, 3) = {m_v, m_v + 4, m_v + 6, m_v + 7, …, 3m_v} and m_v, = v(v ? 1)/6. As a by‐product, we remove the last two undecided cases for the intersection numbers of Kirkman triple system of order 27, this improves the known result provided in [ 2 ]. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 275–289, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10037     

Total colorings of planar graphs with maximum degree at least 8

Planar graphs with maximum degree Δ ⩾ 8 and without 5- or 6-cycles with chords are proved to be (δ + 1)-totally-colorable. This work was supported by Natural Science Foundation of Ministry of Education of Zhejiang Province, China (Grant No. 20070441)     

Edge decompositions and rooted packings of graphs

Let H be a fixed graph. In this paper we consider the problem of edge decomposition of a graph into subgraphs isomorphic to H or $2K_2$ (a 2-edge matching). We give a partial classification of the problems of existence of such decomposition according to the computational complexity. More specifically, for some large class of graphs H we show that this problem is polynomial time solvable and for some other large class of graphs it is NP-complete. These results can be viewed as some edge decomposition analogs of a result by Loebl and Poljak who classified according to the computational complexity the problem of existence of a graph factor with components isomorphic to H or $K_2$. In the proofs of our results we apply so-called rooted packings into graphs which are mutual generalizations of both edge decompositions and factors of graphs.     

Maximal and maximum independent sets in graphs with at most r cycles

Let m(G) denote the number of maximal independent sets of vertices in a graph G and let c(n,r) be the maximum value of m(G) over all connected graphs with n vertices and at most r cycles. A theorem of Griggs, Grinstead, and Guichard gives a formula for c(n,r) when r is large relative to n, while a theorem of Goh, Koh, Sagan, and Vatter does the same when r is small relative to n. We complete the determination of c(n,r) for all n and r and characterize the extremal graphs. Problems for maximum independent sets are also completely resolved. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 283–314, 2006     

Exponentially many maximum genus embeddings and genus embeddings for complete graphs

There are many results on the maximum genus, among which most are written for the existence of values of such embeddings, and few attention has been paid to the estimation of such embeddings and their applications. In this paper we study the number of maximum genus embeddings for a graph and find an exponential lower bound for such numbers. Our results show that in general case, a simple connected graph has exponentially many distinct maximum genus embeddings. In particular, a connected cubic graph G of order n always has at least distinct maximum genus embeddings, where α and m denote, respectively, the number of inner vertices and odd components of an optimal tree T. What surprise us most is that such two extremal embeddings (i.e., the maximum genus embeddings and the genus embeddings) are sometimes closely related with each other. In fact, as applications, we show that for a sufficient large natural number n, there are at least many genus embeddings for complete graph K _n with n ≡ 4, 7, 10 (mod12), where C is a constance depending on the value of n of residue 12. These results improve the bounds obtained by Korzhik and Voss and the methods used here are much simpler and straight. This work was supported by the National Natural Science Foundation of China (Grant No. 10671073), Science and Technology commission of Shanghai Municipality (Grant No. 07XD14011) and Shanghai Leading Academic Discipline Project (Grant No. B407)