Cycles C1 Research Articles

Complete Cycle Embedding in Crossed Cubes with Two-Disjoint-Cycle-Cover Pancyclicity

A graph G is two-disjoint-cycle-cover r-pancyclic if for any integer l satisfying r≤l≤|V(G)|-r, there exist two vertex-disjoint cycles C1 and C2 in G such that the lengths of C1 and C2 are |V(G)|-l and l, respectively, where |V(G)| denotes the total number of vertices in G. In particular, the graph G is two-disjoint-cycle-cover vertex r-pancyclic if for any two distinct vertices u and v of G, there exist two vertex-disjoint cycles C1 and C2 in G such that (i) C1 contains u, (ii) C2 contains v, and (iii) the lengths of C1 and C2 are |V(G)|-l and l, respectively, for any integer l satisfying r≤l≤|V(G)|-r. Moreover, G is two-disjoint-cycle-cover edge r-pancyclic if for any two vertex-disjoint edges (u,v) and (x,y) of G, there exist two vertex-disjoint cycles C1 and C2 in G such that (i) C1 contains (u,v), (ii) C2 contains (x,y), and (iii) the lengths of C1 and C2 are |V(G)|-l and l, respectively, for any integer l satisfying r≤l≤|V(G)|-r. In this paper, we first give Dirac-type sufficient conditions for general graphs to be two-disjoint-cycle-cover vertex/edge 3-pancyclic, and we also prove that the n-dimensional crossed cube CQn is two-disjoint-cycle-cover 4-pancyclic for n≥3, vertex 4-pancyclic for n≥5, and edge 6-pancyclic for n≥5.

Disjoint cycles in hypercubes with prescribed vertices in each cycle

A graph G is spanning r-cyclable of order t if for any r nonempty mutually disjoint vertex subsets A1,A2,…,Ar of G with |A1∪A2∪⋯∪Ar|≤t, there exist r disjoint cycles C1,C2,…,Cr of G such that C1∪C2∪⋯∪Cr spans G, and Ci contains Ai for every i. In this paper, we prove that the n-dimensional hypercube Qn is spanning 2-cyclable of order n−1 for n≥3. Moreover, Qn is spanning k-cyclable of order k if k≤n−1 for n≥2. The spanning r-cyclability of a graph G is the maximum integer t such that G is spanning r-cyclable of order k for k=r,r+1,…,t but is not spanning r-cyclable of order t+1. We also show that the spanning 2-cyclability of Qn is n−1 for n≥3.

A generalization of Littleʼs Theorem on Pfaffian orientations

Little (1975) [12] showed that, in a certain sense, the only minimal non-Pfaffian bipartite matching covered graph is the brace K3,3. Using a stronger notion of minimality than the one used by Little, we show that every minimal non-Pfaffian brick G contains two disjoint odd cycles C1 and C2 such that the subgraph G−V(C1∪C2) has a perfect matching. This implies that the only minimal non-Pfaffian solid matching covered graph is the brace K3,3. (A matching covered graph G is solid if, for any two disjoint odd cycles C1 and C2 of G, the subgraph G−V(C1∪C2) has no perfect matching. Solid matching covered graphs constitute a natural generalization of the class of bipartite graphs, see Carvalho et al., 2004 [5].)

Mutually Independent Hamiltonian Cycle of Burnt Pancake Graphs

Let G=(V,E) be a graph of order n. A Hamiltonian cycle of G is a cycle that contains every vertex in G. Two Hamiltonian cycles C1= and C2= of G are independent if u1=v1 and ui≠vi for 2≤i≤n. A set of Hamiltonian cycles {C1,C2,…,Ck} of G is mutually independent if its elements are pairwise independent. The mutually independent hamiltonicity IHC(G) of a graph G is the maximum integer k such that for any vertex u of G there are k mutually independent Hamiltonian cycles of G starting at u. For the n-dimensional burnt pancake graph Bn, this paper proved that IHC(B2)=1 and IHC(Bn)=n for n≥3.

Circumferences of k-connected graphs involving independence numbers

Let G be a k-connected graph of order n, α: = α(G) the independence number of G, and c(G) the circumference of G. Chvatal and Erdős proved that if α⩽kthen G is hamiltonian. For α⩾k⩾2, Fouquet and Jolivet in 1978 made the conjecture that c(G)⩾k(n+ α − k)/α. Fournier proved that the conjecture is true for α⩽k+ 2 or k = 2 in two different papers. Manoussakis recently proved that the conjecture is true for k = 3. We prove that if G is a k-connected graph, k≥4, of order n and independence number α≥k, then c(G)⩾(k(n+ α − k)/α) − ((k − 3)(k − 4)/2). Consequently, the conjecture of Fouquet and Jolivet holds for k = 4. In addition, we confirm the conjecture for α = k+ 3. Inspired by a result of Kouider, we conjecture that, for every graph G and any two distinct vertices u and v, there is a u − vpath P such that α(G − V(P))⩽α(G) − 1 unless V(G) have a nontrivial partition V1∪V2 satisfying α(G) = α(G[V1]) + α(G[V2]). In this paper, we obtain a partial result regarding this conjecture. Let G be a k-connected graph and k⩾2. While studying the intersection of longest cycles, J. Chen et al. conjectured that, for any two cycles C1 and C2 of G, there are two cycles D1 and D1 such that V(D1)∪V(D2)⊇V(C1)∪V(C2) and |V(D1)∩V(D2)|⩾k. We show that the combination of the above two conjectures implies the conjecture of Fouquet and Jolivet. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:55-76, 2011 © 2011 Wiley Periodicals, Inc.

On 2-factors with cycles containing specified vertices in a bipartite graph

Let k≥1 be an integer and G=(V 1,V 2;E) a bipartite graph with |V 1|=|V 2|=n such that n≥2k+2. Our result is as follows: If $d(x)+d(y)\geq \lceil\frac{4n+k}{3}\rceil$ for any nonadjacent vertices x∈V 1 and y∈V 2, then for any k distinct vertices z 1,…,z k , G contains a 2-factor with k+1 cycles C 1,…,C k+1 such that z i ∈V(C i ) and l(C i )=4 for each i∈{1,…,k}.

Weakly Complementary Cycles in 3-Connected Multipartite Tournaments

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 C1 and C2 such that V (D) = V (C1) ( V (C2), and a multipartite tournament D is called weakly cycle complementary if there exist two vertex disjoint cycles C1 and C2 such that V (C1) ( V (C2) 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 jV (T)j i t for all 3 • tjV (T)j=2. Recently, Volkmann (8) proved that each regular multipartite tournament D of order jV (D)j ‚ 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 ‚ 3 that are weakly cycle complementary. 1. Terminology In this paper all digraphs are flnite without loops and multiple arcs. The vertex set and the arc set of a digraph D are denoted by V (D) and E(D), respectively. If xy is an arc of a digraph D, then we write x ! y and say x dominates y, and if X and Y are two disjoint vertex sets or subdigraphs of D such that every vertex of X dominates every vertex of Y , then we say that X dominates Y , denoted by X ! Y. Furthermore, X ; Y denotes the fact that there is no arc leading from Y to X. If D is a digraph, then the out-neighborhood N + D (x) = N + (x) of a vertex x is the set of vertices dominated by x and the in-neighborhood N i D (x) = N i (x) is the set of vertices dominating x. Therefore, if the arc xy 2 E(D) exists, then y is an outer neighbor of x and x is an inner neighbor of y. The numbers d + (x) = d + (x) = jN + (x)j and d i (x) = d i (x) = jN i (x)j are called the outdegree and the indegree of x, respectively. Furthermore, the numbers - + D = - + = minfd + (x)jx 2 V (D)g and - i D = - i = minfd i (x)jx 2 V (D)g are the minimum outdegree and the minimum

Characterization of edge‐colored complete graphs with properly colored Hamilton paths

AbstractAn edge‐colored graph H is properly colored if no two adjacent edges of H have the same color. In 1997, J. Bang‐Jensen and G. Gutin conjectured that an edge‐colored complete graph G has a properly colored Hamilton path if and only if G has a spanning subgraph consisting of a properly colored path C0 and a (possibly empty) collection of properly colored cycles C1,C2,…, Cd such that $V (C_i) \cap {V(C}_j) =\emptyset$ provided $0 \le i < j \le d$. We prove this conjecture. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 333–346, 2006

On 2–Factors with Prescribed Properties in a Bipartite Graph

Liu and Yan gave the degree condition for a balanced bipartite graph G = (V 1, V 2;E) to have k vertex–disjoint quadrilaterals containing any given k independent edges e 1, . . . , e k of G, respectively. They also conjectured that for any k independent edges e 1, . . . , e k of G, G has a 2–factor with k cycles C 1,C 2, . . . , C k with respect to {e 1, e 2, . . . , e k } such that k–1 of them are quadrilaterals. In this paper, we prove this conjecture.

Vertex-Disjoint Cycles Containing Specified Vertices in a Graph

Let k ≥ 2 and n ≥ 1 be integers, and let G be a graph of order n with minimum degree at least k + 1. Let v1, v2, …,vk be k distinct vertices of G, and suppose that there exist vertex-disjoint cycles C1, C2, …, Ck in G such that and vi ∊ V (Ci) for each 1 ≤ i ≤ k. Suppose further that the minimum value of the sum of the degrees of two nonadjacent distinct vertices is greater than or equal to n. Under these assumptions, we show that there exist vertex-disjoint cycles D1, D2, …, Dk such that vi ∊ V (Di) for each 1 ≤ i ≤ k and .

Approximating the Minmax Rooted-Subtree Cover Problem

Let G = (V, E) be a connected graph such that each edge e ∈ E and each vertex v ∈ V are weighted by nonnegative reals w(e) and h(v), respectively. Let r be a vertex designated as a root, and p be a positive integer. The minmax rooted-subtree cover problem (MRSC) asks to find a partition χ = {X1, X2, ..., Xp} of V and a set of p subtrees T1, T2, ..., Tp such that each Ti contains Xi ∪ {r} so as to minimize the maximum cost of the subtrees, where the cost of Ti is defined to be the sum of the weights of edges in Ti and the weights of vertices in Xi. Similarly, the minmax rooted-cycle cover problem (MRCC) asks to find a partition χ = {X1, X2, ..., Xp} of V and a set of p cycles C1, C2, ..., Cp such that Ci contains Xi ∪ {r} so as to minimize the maximum cost of the cycles, where the cost of Ci is defined analogously with the MRSC. In this paper, we first propose a (3-2/(p + 1))-approximation algorithm to the MRSC with a general graph G, and we then give a (6 - 4/(p + 1))-approximation algorithm to the MRCC with a metric (G, w).

Degree-Sum Conditions for Graphs to Have 2-Factors with Cycles Through Specified Vertices

Let k≥2 and n≥1 be integers, let G be a graph of order n with minimum degree at least k + 1. Let υ1,υ2,…,υk be k distinct vertices of G, and suppose that there exist k vertex disjoint cycles C1,…,Ck in G such that υi∈V(Ci) for each 1≤i≤k. Suppose further that the minimum value of the sum of the degrees of two nonadjacent distinct vertices is greater than or equal to n+k−43. Under these assumptions, we show that there is a 2-factor of G with k cycles D1,D2,…,Dk such that υi∈V(Di) for each 1≤i≤k.

Coloring Locally Bipartite Graphs on Surfaces

It is proved that there is a function f:N→N such that the following holds. Let G be a graph embedded in a surface of Euler genus g with all faces of even size and with edge-width ⩾f(g). Then (i)If every contractible 4-cycle of G is facial and there is a face of size >4, then G is 3-colorable. (ii)If G is a quadrangulation, then G is not 3-colorable if and only if there exist disjoint surface separating cycles C1, …, Cg such that, after cutting along C1, …, Cg, we obtain a sphere with g holes and g Möbius strips, an odd number of which is nonbipartite. If embeddings of graphs are represented combinatorially by rotation systems and signatures [5], then the condition in (ii) is satisfied if and only if the geometric dual of G has an odd number of edges with negative signature.

Proof of a conjecture on cycles in a bipartite graph

It was conjectured in [Wang, to appear in The Australasian Journal of Combinatorics] that, for each integer k ≥ 2, there exists M(k) such that if G= (V1, V2, E) is a bipartite graph with |V1| = |V2| = n ≥ M(k) and d(x) + d(y) ≥ n + k for each pair of nonadjacent vertices x and y of G with x ∈ V1 and y ∈ V2, then, for any k independent edges e1, …, ek of G, there exist k vertex-disjoint cycles C1, …, Ck in G such that ei ∈ E(Ci for all i ∈ {1, …, k} and V(C1 ∪ ··· ∪ Ck) = V(G). This conjecture is also verified for k = 2, 3 in [Wang, to appear; Wang, manuscript]. In this article, we prove this conjecture to be true if n ≥ 3k, i.e., M(k) ≤ 3k. We will also show that, if n ≥ 3k, then for any k independent edges e1, …, ek of G, there exist k vertex-disjoint cycles C1, …, Ck of length at most 6 in G such that ei ∈ E(Ci) for all i ∈ {1, …, k}. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 333–343, 1999

Covering a graph with cycles passing through given edges

We propose a conjecture: for each integer k ≥ 2, there exists N(k) such that if G is a graph of order n ≥ N(k) and d(x) + d(y) ≥ n + 2k - 2 for each pair of non-adjacent vertices x and y of G, then for any k independent edges e1, …, ek of G, there exist k vertex-disjoint cycles C1, …, Ck in G such that ei ∈ E(Ci) for all i ∈ {1, …, k} and V(C1 ∪ ···∪ Ck) = V(G). If this conjecture is true, the condition on the degrees of G is sharp. We prove this conjecture for the case k = 2 in the paper. © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 105–109, 1997

Paths, cycles, and arc‐connectivity in digraphs

AbstractIn this paper we prove the following theorem: Let D be a k‐arcconnected digraph (multiple arcs allowed). If x is a vertex of D and / is an integer with / ≤ k, then for any / disjoint arc pairs {f1, g1}, ⃛, {f1, g1}, where f1, ⃛, f1 are arcs with head at x and g1, ⃛, g1 are arcs with tail at x, there exist in D / arc‐disjoint cycles C1, ⃛, C1 such that {fi, gi} ⊆ E(Ci) for each i (E(Ci) denotes the arc set of Ci) and such that D ‐ ∪ E(Ci) is (k ‐ 1)‐arc‐connected. Several interesting results are deduced from this theorem. Our results generalize the early works of Mader (“On a Property of n‐Edge‐Connected Digraphs,” Combinatorica, vol. 1 [1981], pp. 385‐386) and Shiloach (“Edge‐Disjoint Branching in Directed Multigraphs,” Information Processing Letters, vol. 8 [1979], pp. 24‐27). An extension of Mader's theorem about admissible liftings of digraphs is also obtained in this paper. © 1995 John Wiley & Sons, Inc.