Partition and coloring of a graph

·A3-connected graph of order at least t can be partitioned into connected subgraphs of order at least t and at most 2t-1。Using this result，we can show that a3-connected graph of order at least t contains a connected subgraph of order t whose degree sum is at most 8t-1。·For a3-page book embedding of a graph of order n and size m，the order of the minimum number of edge-crossings orer the spine is O(M Log I D 1 N I D 1)。·Let G be a k-connected graph of order p，p-n is even，αis a real number with i D71(/)2个D 7≤α≤1，and suppose|Na(A)|&gt；α(p-2k n-2)k for any indegredant subset A of G with|A|=[α(k-n2)]。Then G is n-factor-critical.·Let G be a balanced bipartite graph of order 2n，K≥2，n≥2k..SupposeσI D21 I D 2，(G1)≧max{n k，[I D 72 n-1(/)3 I D 7]2 k}orδ(G)≧max{[I D 7 n k(/)2 I D 7]，[I D 72 n 4 k(/)5 I D 7]}。Then for any independent edges。E I D 21 I D 2，--，E D 2 k D 2，G can be partitioned into disjoint cycles C I D 21 D 2，-，C D 2 k D 2 satisfying e D 2 i i D 2εE(G I D 2 I D 2)1≤I≤k..

・A 3-connected graph of order at least t can be partitioned into connected subgraphs of order at least t and at most 2t-1. Using this result, we can show that a 3-connected graph of order at least t contains a connected subgraph of order t whose degree sum is at most 8t-1.・For a 3-page book embedding of a graph of order n and size m, the order of the minimum number of edge-crossings orer the spine is O (m logィイD1nィエD1).・Let G be a k-connected graph of order p, p-n is even, α is a real number with ィイD71(/)2ィエD7≦α≦1, and suppose |Na(a)|>α(p-2k+n-2)+k for any indegredant subset A of G with |A|=[α(k-n+2)]. Then G is n-factor-critical.・Let G be a balanced bipartite graph of order 2n, K≧2, n≧2k.. Suppose σィイD21ィエD2,(G1)≧ max {n+k, [ィイD72n-1(/)3ィエD7]+2k} or δ(G)≧ max {[ィイD7n+k(/)2ィエD7],[ィイD72n+4k(/)5ィエD7]}. Then for any independent edges. EィイD21ィエD2,---, eィイD2kィエD2, G can be partitioned into disjoint cycles CィイD21ィエD2,----, CィイD2kィエD2 satisfying eィイD2iィエD2 εE (GィイD2iィエD2) 1≦I≦k..