• Journal of applied mathematics & informatics
• /
• v.27 no.3_4
• /
• pp.809-817
• /
• 2009

We call a cycle C be a small cycle if the length of C equals to 3 or 4. In this paper, we obtain two sufficient conditions to ensure the existence of vertex-disjoint small cycles in graph and propose several problems.

• Journal of Information Processing Systems
• /
• v.7 no.1
• /
• pp.75-84
• /
• 2011

The enhanced pyramid graph was recently proposed as an interconnection network model in parallel processing for maximizing regularity in pyramid networks. We prove that there are two edge-disjoint Hamiltonian cycles in the enhanced pyramid networks. This investigation demonstrates its superior property in edge fault tolerance. This result is optimal in the sense that the minimum degree of the graph is only four.

• Journal of the Korean Mathematical Society
• /
• v.51 no.5
• /
• pp.919-940
• /
• 2014

We conjecture that if $k{\geq}2$ is an integer and G is a graph of order n with minimum degree at least (n+2k)/2, then for any k independent edges $e_1$, ${\cdots}$, $e_k$ in G and for any integer partition $n=n_1+{\cdots}+n_k$ with $n_i{\geq}4(1{\leq}i{\leq}k)$, G has k disjoint cycles $C_1$, ${\cdots}$, $C_k$ of orders $n_1$, ${\cdots}$, $n_k$, respectively, such that $C_i$ passes through $e_i$ for all $1{\leq}i{\leq}k$. We show that this conjecture is true for the case k = 2. The minimum degree condition is sharp in general.

• Bulletin of the Korean Mathematical Society
• /
• v.52 no.1
• /
• pp.287-299
• /
• 2015

A theta graph is the union of three internally disjoint paths that have the same two distinct end vertices. We show that every graph of order $n{\geq}12$ and size at least ${\lfloor}\frac{11n-18}{2}{\rfloor}$ contains three disjoint theta graphs. As a corollary, every graph of order $n{\geq}12$ and size at least ${\lfloor}\frac{11n-18}{2}{\rfloor}$ contains three disjoint cycles of even length.

• Journal of KIISE:Computer Systems and Theory
• /
• v.26 no.8
• /
• pp.999-1007
• /
• 1999

이 논문은 재귀원형군 G(2^m , 2^k )를 그래프 이론적 관점에서 고찰하고 정점이 서로소인 사이클과 그래프 invariant에 관한 위상 특성을 제시한다. 재귀원형군은 1 에서 제안된 다중 컴퓨터의 연결망 구조이다. 재귀원형군 {{{{G(2^m , 2^k )가 길이 사이클을 가질 필요 충분 조건을 구하고, 이 조건하에서 G(2^m , 2^k )는 가능한 최대 개수의 정점이 서로소이고 길이가l`인 사이클을 가짐을 보인다. 그리고 정점 및 에지 채색, 최대 클릭, 독립 집합 및 정점 커버에 대한 그래프 invariant를 분석한다.Abstract In this paper, we investigate recursive circulant G(2^m , 2^k ) from the graph theory point of view and present topological properties of G(2^m , 2^k ) concerned with vertex-disjoint cycles and graph invariants. Recursive circulant is an interconnection structure for multicomputer networks proposed in 1 . A necessary and sufficient condition for recursive circulant {{{{G(2^m , 2^k ) to have a cycle of lengthl` is derived. Under the condition, we show that G(2^m , 2^k ) has the maximum possible number of vertex-disjoint cycles of length l`. We analyze graph invariants on vertex and edge coloring, maximum clique, independent set and vertex cover.

• East Asian mathematical journal
• /
• v.30 no.3
• /
• pp.311-319
• /
• 2014

It is shown that, for an even positive integer m with $m{\geq}4$ and arbitrary positive integer k and t, the complete multipartite graph $K_{km+1(2t)}$ can be decomposed into edge-disjoint gregarious m-cycles in such a way that the decomposition is circulant.

• Kyungpook Mathematical Journal
• /
• v.48 no.2
• /
• pp.287-302
• /
• 2008

The vertex set of a digraph D is denoted by V (D). A c-partite tournament is an orientation of a complete c-partite graph. A digraph D is called cycle complementary if there exist two vertex disjoint cycles $C_1$ and $C_2$ such that V(D) = $V(C_1)\;{\cup}\;V(C_2)$, and a multipartite tournament D is called weakly cycle complementary if there exist two vertex disjoint cycles $C_1$ and $C_2$ such that $V(C_1)\;{\cup}\;V(C_2)$ contains vertices of all partite sets of D. The problem of complementary cycles in 2-connected tournaments was completely solved by Reid [4] in 1985 and Z. Song [5] in 1993. They proved that every 2-connected tournament T on at least 8 vertices has complementary cycles of length t and ${\mid}V(T)\mid$ - t for all $3\;{\leq}\;t\;{\leq}\;{\mid}V(T)\mid/2$. Recently, Volkmann [8] proved that each regular multipartite tournament D of order ${\mid}V(D)\mid\;\geq\;8$ is cycle complementary. In this article, we analyze multipartite tournaments that are weakly cycle complementary. Especially, we will characterize all 3-connected c-partite tournaments with $c\;\geq\;3$ that are weakly cycle complementary.

• East Asian mathematical journal
• /
• v.29 no.3
• /
• pp.337-347
• /
• 2013

For an even integer m at least 4 and any positive integer $t$, it is shown that the complete equipartite graph $K_{km(2t)}$ can be decomposed into edge-disjoint gregarious m-cycles for any positive integer ${\kappa}$ under the condition satisfying ${\frac{{(m-1)}^2+3}{4m}}$ < ${\kappa}$. Here it will be called a gregarious cycle if the cycle has at most one vertex from each partite set.

• Bulletin of the Korean Mathematical Society
• /
• v.49 no.1
• /
• pp.145-154
• /
• 2012

Let G be a planar graph with maximum degree $\Delta$. The linear 2-arboricity $la_2$(G) of G is the least integer k such that G can be partitioned into k edge-disjoint forests, whose component trees are paths of length at most 2. In this paper, we prove that (1) $la_2(G){\leq}{\lceil}\frac{\Delta}{2}\rceil+8$ if G has no adjacent 3-cycles; (2) $la_2(G){\leq}{\lceil}\frac{\Delta}{2}\rceil+10$ if G has no adjacent 4-cycles; (3) $la_2(G){\leq}{\lceil}\frac{\Delta}{2}\rceil+6$ if any 3-cycle is not adjacent to a 4-cycle of G.

• Communications of the Korean Mathematical Society
• /
• v.20 no.3
• /
• pp.603-610
• /
• 2005

A k-sized balanced sampling plan excluding contiguous units of order v and index denoted by $BSEC(v,\;k,\;{\lambda})$, is said to be bicyclic if it admits an automorphism consisting of two disjoint cycles of length ~. In this paper, we obtain a necessary and sufficient condition for the existence of bicyclic BSEC(v, 3, 2)s.