Equitable List Coloring of Graphs with Bounded Degree

A graph G is equitably k‐choosable if for every k‐list assignment L there exists an L‐coloring of G such that every color class has at most vertices. We prove results toward the conjecture that every graph with maximum degree at most r is equitably ‐choosable. In particular, we confirm the conjecture for and show that every graph with maximum degree at most r and at least r³ vertices is equitably ‐choosable. Our proofs yield polynomial algorithms for corresponding equitable list colorings.     

若干联图的均匀全染色

图$G(V,E)$的全色数 $\chi_{t}(G)$就是将$V\bigcup E$分成彼此不相交的全独立分割集的最小个数。 如果任何两个$V\bigcup E$的全独立分割集的元素数目相差不超过1，那么 $V \bigcup E$的全独立分割集的最小个数就称为图$G$的均匀全色数，记为$\chi_{et}(G)$。 在本文中我们给出了当 $m \geq n \geq 3$ 时 $W_m\bigvee K_n$，$F_m \bigvee K_n$及$S_m \bigvee K_n$ 的均匀全色数.     

最大度为4的图的无圈列表边染色

对于图G=(V(G),E(G)),如果一个映射φ:E(G)→{1,2,…,k},使得G中任意相邻的两边e₁,e₂满足φ(e₁)≠φ(e₂),并且G中不含有双色圈,则称φ为G的一个无圈边染色.对于给定的列表分配L={L(e)|e∈E(G)},如果存在图G的一个无圈边染色φ,使得对于任意边e∈E(G),均有φ(e)∈L(e),则称染色φ为G的一个无圈L-边染色.如果对于任意的列表分配L,当对所有的边e∈E(G)满足|L(e)|≥k时,图G均存在无圈L-边染色,那么称G是无圈k-边可选的.使图G无圈k-边可选的最小的正整数k,称为G的无圈列表边色数,用a’_l(G)表示.本文证明了对于最大度△≤4的连通图G,如果|E(G)|≤2|V(G)|-1,则a’_l(G)≤6,扩展了Basavaraju和Chandran文[J.Graph Theory,2009,61(3):192-209]的结果.     

一些联图的均匀全染色

对一个正常的全染色满足各种颜色所染元素数(点或边)相差不超过1时,称为均匀全染色,其所用最少染色数称为均匀全色数.本文证明了图在若干情况下的均匀全色数定理,得到C_m∨S_nC_m∨ F_n和C∨W_n的均匀全色数.     

关于一类二部图的均匀邻点可区别全染色

若图的邻点可区别全染色的各色所染元素数之差不超过1,则称该染色法为图的均匀邻点可区别全染色,而所用的最少颜色数称为该图的均匀邻点可区别全色数.本文给出了一类二部图的均匀邻点可区别全染色数.     

若干倍图的均匀全染色(英文)

如果图G的一个正常全染色满足任意两种颜色所染元素(点或边)数目相差不超过1,则称为G的均匀全染色,其所用最少染色数称为均匀全色数.本文得到了星、扇和轮的倍图的均匀全色数.     

若干Mycielski图的均匀染色

如果图G的一个正常顶点染色满足任两个色类中的顶点数相差不超过1,则称为G的均匀染色.研究了一些Mycielski图的均匀染色,给出了路、圈、完全图和广义星图的Mycielski图的均匀色数.     

Equitable Coloring of Complete r-partite Graphs

A graph is equitably k-colorable if its vertices can be partitioned into k independent sets of as near equal sizes as possible. In this paper, we determine a sufficient and necessary condition for which a complete r-partite graph is equitably k-colorable. From this result, we can provide another way to prove some previous results.     

最大度为8不含特定子图的平面图的全染色

全染色是对图G的顶点和边同时进行正常染色,至少要用Δ+1个色才能对图G进行正常全染色.本文运用权转移的方法,证明了最大度为8的不含特定子图的简单平面图是9-全可染的.     

P_n~k的均匀全染色

设G(V,E)是一个简单图,f是G的一个k-正常全染色,若f满足||Vi∪Ei|-|Vj∪Ej||≤1(i≠j),其中Vi∪Ei={v|f(v)=i}∪{e|f(e)=i},则称f为G的k-均匀全染色,简记为k-ETC.并称eχT(G)=min{k|G存在k-均匀全染色}为G的均匀全染色数.本文将通过很好的全染色方法得到eχT(Pkn)=5(n≥2k+1),并证明了对Pkn,[5]中猜想是正确的.     

Equitable Coloring of Three Classes of 1-planar Graphs

A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. A plane graph with near-independent crossings or independent crossings, say NIC-planar graph or IC-planar graph, is a 1-planar graph with the restriction that for any two crossings the four crossed edges are incident with at most one common vertex or no common vertices, respectively. In this paper, we prove that each 1-planar graph, NIC-planar graph or IC-planar graph with maximum degree Δ at least 15, 13 or 12 has an equitable Δ-coloring, respectively. This verifies the well-known Chen-Lih-Wu Conjecture for three classes of 1-planar graphs and improves some known results.     

Pkn的均匀全染色

设G(V,E)是一个简单图,f是G的一个k-正常全染色,若f满足||Vi∪Ei|-|Vj∪Ej||≤1(i≠j),其中Vi∪Ei={v|f(v)=i}∪{e|f(e)=i},则称f为G的k-均匀全染色,简记为k-ETC.并称χeT(G)=min{k|G存在k-均匀全染色}为G的均匀全染色数.本文将通过很好的全染色方法得到χeT(Pkn)=5(n≥2k 1),并证明了对Pkn,[5]中猜想是正确的.     

Induced-Universal Graphs for Graphs with Bounded Maximum Degree

For a family \(\mathcal {F}\) of graphs, a graph U is induced-universal for \({\mathcal{F}}\) if every graph in \({\mathcal{F}}\) is an induced subgraph of U. We give a construction for an induced-universal graph for the family of graphs on n vertices with degree at most r, which has \(Cn^{\lfloor (r+1)/2\rfloor}\) vertices and \(Dn^{2\lfloor (r+1)/2\rfloor -1}\) edges, where C and D are constants depending only on r. This construction is nearly optimal when r is even in that such an induced-universal graph must have at least cn ^r/2 vertices for some c depending only on r.Our construction is explicit in that no probabilistic tools are needed to show that the graph exists or that a given graph is induced-universal. The construction also extends to multigraphs and directed graphs with bounded degree.     

外平面图的全染色与列表全染色

本文证明了,如果G是满足条件Δ(G)≥4的外平面图,则x_T~L(G)=Δ(G) 1,同时对Δ(G)=3给出了XT(G)=Δ(G) 1的简短的新证明,从而蕴含Δ(G)≥3时,XT(G)=Δ(G) 1,其中XT(G)是G的点边全色数,x_T~L(G)是G的点边列表全色数。     

关于图的均匀全色数分类

对一个正常的全染色满足各种颜色所染元素数(点或边)相差不超过1时,称为均匀全染色,其所用最少染色数称为均匀全色数.将图按均匀全色数分类,证明了简单图在若干情况下的均匀全色数定理,得到了一些联图的均匀全色数.     

关于多重联图的均匀全染色

对一个正常的全染色满足各种颜色所染元素数(点或边)相差不超过1时,称为均匀全染色,其所用最少染色数称为均匀全色数．本文证明了关于多重联图的若干情况下的均匀全色数定理,得到了若干特殊多重联图的均匀全色数．     

一些联图的均匀全染色

对-个正常的全染色满足各种颜色所染元素数(点或边)相差不超过1时,称为均匀全染色,其所用最少染色数称为均匀全色数.本文证明了图在若干情况下的均匀全色数定理,得到了Cm VSn,CmVFn和CmVWn的均匀全色数.     

Total-coloring of Sparse Graphs with Maximum Degree 6

Acta Mathematicae Applicatae Sinica, English Series - Given a graph G = (V, E) and a positive integer k, a k-total-coloring of G is a mapping φ: V?E → {1, 2, ?, k} such that...