Heavy cycles in k-connected weighted graphs with large weighted degree sums

A weighted graph is one in which every edge e is assigned a nonnegative number w(e), called the weight of e. The weight of a cycle is defined as the sum of the weights of its edges. The weighted degree of a vertex is the sum of the weights of the edges incident with it. In this paper, we prove that: Let G be a k-connected weighted graph with k?2. Then G contains either a Hamilton cycle or a cycle of weight at least 2m/(k+1), if G satisfies the following conditions: (1) The weighted degree sum of any k+1 pairwise nonadjacent vertices is at least m; (2) In each induced claw and each induced modified claw of G, all edges have the same weight. This generalizes an early result of Enomoto et al. on the existence of heavy cycles in k-connected weighted graphs.     

Sur les arborescences dans un graphe oriente

The main result of this paper is the following theorem: Let G = (X,E) be a digraph without loops or multiple edges, |X| ?3, and h be an integer ?1, if G contains a spanning arborescence and if d⁺_G(x)+d^?_G(x)+d^?_G(y)+d^?_G(y)? 2|X |?2h?1 for all x, y?X, x ≠ y, non adjacent in G, then G contains a spanning arborescence with ?h terminal vertices. A strengthening of Gallai-Milgram's theorem is also proved.     

Complete Multipartite Graphs and the Relaxed Coloring Game

Let k be a positive integer, d be a nonnegative integer, and G be a finite graph. Two players, Alice and Bob, play a game on G by coloring the uncolored vertices with colors from a set X of k colors. At all times, the subgraph induced by a color class must have maximum degree at most d. Alice wins the game if all vertices are eventually colored; otherwise, Bob wins. The least k such that Alice has a winning strategy is called the d-relaxed game chromatic number of G, denoted ?? _g ^d (G). It is known that there exist graphs such that ?? _g ⁰(G)?=?3, but ?? _g ¹(G)?>?3. We will show that for all positive integers m, there exists a complete multipartite graph G such that m?????? _g ⁰(G)?<??? _g ¹(G).     

Hamiltonian cycles and dominating cycles passing through a linear forest

Let G be an (m+2)-graph on n vertices, and F be a linear forest in G with |E(F)|=m and ω₁(F)=s, where ω₁(F) is the number of components of order one in F. We denote by σ₃(G) the minimum value of the degree sum of three vertices which are pairwise non-adjacent. In this paper, we give several σ₃ conditions for a dominating cycle or a hamiltonian cycle passing through a linear forest. We first prove that if σ₃(G)≥n+2m+2+max{s−3,0}, then every longest cycle passing through F is dominating. Using this result, we prove that if σ₃(G)≥n+κ(G)+2m−1 then G contains a hamiltonian cycle passing through F. As a corollary, we obtain a result that if G is a 3-connected graph and σ₃(G)≥n+κ(G)+2, then G is hamiltonian-connected.     

On a Spanning Tree with Specified Leaves

Let k ≥ 2 be an integer. We show that if G is a (k + 1)-connected graph and each pair of nonadjacent vertices in G has degree sum at least |G| + 1, then for each subset S of V(G) with |S| = k, G has a spanning tree such that S is the set of endvertices. This result generalizes Ore’s theorem which guarantees the existence of a Hamilton path connecting any two vertices. Dedicated to Professor Hikoe Enomoto on his 60th birthday.     

On the graph of large distances

For a setS of points in the plane, letd ₁>d ₂>... denote the different distances determined byS. Consider the graphG(S, k) whose vertices are the elements ofS, and two are joined by an edge iff their distance is at leastd _k . It is proved that the chromatic number ofG(S, k) is at most 7 if |S|constk ². IfS consists of the vertices of a convex polygon and |S|constk ², then the chromatic number ofG(S, k) is at most 3. Both bounds are best possible. IfS consists of the vertices of a convex polygon thenG(S, k) has a vertex of degree at most 3k – 1. This implies that in this case the chromatic number ofG(S, k) is at most 3k. The best bound here is probably 2k+1, which is tight for the regular (2k+1)-gon.     

Disjoint matchings of graphs

In this paper, we prove a generalization of the familiar marriage theorem. One way of stating the marriage theorem is: Let G be a bipartite graph, with parts S₁ and S₂. If A ? S₁ and F(A) ? S₂ is the set of neighbors of points in A, then a matching of G exists if and only if Σ_x∈S2 min(1, | F^?1(x) ∩ A |) ≥ | A | for each A ? S₁. Our theorem is that k disjoint matchings of G exist if and only Σ_x∈S2 min (k, | F^?1(x) ∩ A |) ≥ k | A | for each A ? S₁.     

The equivariant topology of stable Kneser graphs

The stable Kneser graph SG_n,k, n?1, k?0, introduced by Schrijver (1978) [19], is a vertex critical graph with chromatic number k+2, its vertices are certain subsets of a set of cardinality m=2n+k. Björner and de Longueville (2003) [5] have shown that its box complex is homotopy equivalent to a sphere, Hom(K₂,SG_n,k)?Sk. The dihedral group D2m acts canonically on SG_n,k, the group C₂ with 2 elements acts on K₂. We almost determine the (C₂×D2m)-homotopy type of Hom(K₂,SG_n,k) and use this to prove the following results.The graphs SG_2s,4 are homotopy test graphs, i.e. for every graph H and r?0 such that Hom(SG_2s,4,H) is (r−1)-connected, the chromatic number χ(H) is at least r+6.If k∉{0,1,2,4,8} and n?N(k) then SG_n,k is not a homotopy test graph, i.e. there are a graph G and an r?1 such that Hom(SG_n,k,G) is (r−1)-connected and χ(G)<r+k+2.     

On a conjecture of bondy

Bondy conjectured [1] that: if G is a k-connected graph, where k ≥ 2, such that the degree-sum of any k + 1 independent vertices is at least m, then G contains a cycle of length at least: $\text{Min}\text{(}\text{2m}\text{(k + 1)}\text{, n)}$ (n denotes the order of G). We prove here that this result is true.     

Degree sum conditions for oriented forests in digraphs

Let F be an oriented forest with n vertices and m arcs and D be a digraph without loops and multiple arcs. In this note we prove that D contains a subdigraph isomorphic to F if D has at least n vertices and min{d⁺(u)+d⁺(v),d⁻(u)+d⁻(v),d⁺(u)+d⁻(v)}≥2m−1 for every pair of vertices u,v∈V(D) with uv∉A(D). This is a common generalization of two results of Babu and Diwan, one on the existence of forests in graphs under a degree sum condition and the other on the existence of oriented forests in digraphs under a minimum degree condition.     

On cycles in regular 3-partite 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. In 1991, Jian-zhong Wang conjectured that every arc of a regular 3-partite tournament D is contained in directed cycles of all lengths 3,6,9,…,|V(D)|. This conjecture is not valid, because for each integer t with 3?t?|V(D)|, there exists an infinite family of regular 3-partite tournaments D such that at least one arc of D is not contained in a directed cycle of length t.In this paper, we prove that every arc of a regular 3-partite tournament with at least nine vertices is contained in a directed cycle of length m, m+1, or m+2 for 3?m?5, and we conjecture that every arc of a regular 3-partite tournament is contained in a directed cycle of length m, (m+1), or (m+2) for each m∈{3,4,…,|V(D)|-2}.It is known that every regular 3-partite tournament D with at least six vertices contains directed cycles of lengths 3, |V(D)|-3, and |V(D)|. We show that every regular 3-partite tournament D with at least six vertices also has a directed cycle of length 6, and we conjecture that each such 3-partite tournament contains cycles of all lengths 3,6,9,…,|V(D)|.     

Spanning Connectivity of the Power of a Graph and Hamilton-Connected Index of a Graph

Let G = (V, E) be a connected graph. The hamiltonian index h(G) (Hamilton-connected index hc(G)) of G is the least nonnegative integer k for which the iterated line graph L ^k (G) is hamiltonian (Hamilton-connected). In this paper we show the following. (a) If |V(G)| ≥ k + 1 ≥ 4, then in G ^k , for any pair of distinct vertices {u, v}, there exists k internally disjoint (u, v)-paths that contains all vertices of G; (b) for a tree T,  h(T) ≤ hc(T) ≤ h(T) + 1, and for a unicyclic graph G,  h(G) ≤ hc(G) ≤ max{h(G) + 1, k′ + 1}, where k′ is the length of a longest path with all vertices on the cycle such that the two ends of it are of degree at least 3 and all internal vertices are of degree 2; (c) we also characterize the trees and unicyclic graphs G for which hc(G) = h(G) + 1.     

Generalization of matching extensions in graphs (III)

Proposing them as a general framework, Liu and Yu (2001) [6] introduced (n,k,d)-graphs to unify the concepts of deficiency of matchings, n-factor-criticality and k-extendability. Let G be a graph and let n,k and d be non-negative integers such that n+2k+d+2?|V(G)| and |V(G)|−n−d is even. If on deleting any n vertices from G the remaining subgraph H of G contains a k-matching and each k-matching can be extended to a defect-d matching in H, then G is called an (n,k,d)-graph. In this paper, we obtain more properties of (n,k,d)-graphs, in particular the recursive relations of (n,k,d)-graphs for distinct parameters n,k and d. Moreover, we provide a characterization for maximal non-(n,k,d)-graphs.     

F-Sets in graphs

A subset S of the vertex set of a graph G is called an F-set if every α?Γ(G), the automorphism group of G, is completely specified by specifying the images under α of all the points of S, and S has a minimum number of points. The number of points, k(G), in an F-set is an invariant of G, whose properties are studied in this paper. For a finite group Γ we define k(Γ) = max{k(G) | Γ(G) = Γ}. Graphs with a given Abelian group and given k-value (k ≤ k(Γ)) have been constructed. Graphs with a given group and k-value 1 are constructed which give simple proofs to the theorems of Frucht and Bouwer on the existence of graphs with given abstract/permutation groups.     

On degree sum conditions for long cycles and cycles through specified vertices

Let G be a graph. For S⊂V(G), let Δ_k(S) denote the maximum value of the degree sums of the subsets of S of order k. In this paper, we prove the following two results. (1) Let G be a 2-connected graph. If Δ₂(S)≥d for every independent set S of order κ(G)+1, then G has a cycle of length at least min{d,|V(G)|}. (2) Let G be a 2-connected graph and X a subset of V(G). If Δ₂(S)≥|V(G)| for every independent set S of order κ(X)+1 in G[X], then G has a cycle that includes every vertex of X. This suggests that the degree sum of nonadjacent two vertices is important for guaranteeing the existence of these cycles.     

A generalization of Fan’s results: Distribution of cycle lengths in graphs

Fan [G. Fan, Distribution of cycle lengths in graphs, J. Combin. Theory Ser. B 84 (2002) 187-202] proved that if G is a graph with minimum degree δ(G)≥3k for any positive integer k, then G contains k+1 cycles C₀,C₁,…,C_k such that k+1<|E(C₀)|<|E(C₁)|<?<|E(C_k)|, |E(C_i)−E(C_i−1)|=2, 1≤i≤k−1, and 1≤|E(C_k)|−|E(C_k−1)|≤2, and furthermore, if δ(G)≥3k+1, then |E(C_k)|−|E(C_k−1)|=2. In this paper, we generalize Fan’s result, and show that if we let G be a graph with minimum degree δ(G)≥3, for any positive integer k (if k≥2, then δ(G)≥4), if d_G(u)+d_G(v)≥6k−1 for every pair of adjacent vertices u,v∈V(G), then G contains k+1 cycles C₀,C₁,…,C_k such that k+1<|E(C₀)|<|E(C₁)|<?<|E(C_k)|, |E(C_i)−E(C_i−1)|=2, 1≤i≤k−1, and 1≤|E(C_k)|−|E(C_k−1)|≤2, and furthermore, if d_G(u)+d_G(v)≥6k+1, then |E(C_k)|−|E(C_k−1)|=2.     

An approximate version of the Loebl-Komlós-Sós conjecture

Loebl, Komlós, and Sós conjectured that if at least half of the vertices of a graph G have degree at least some k∈N, then every tree with at most k edges is a subgraph of G. Our main result is an approximate version of this conjecture for large enough n=|V(G)|, assumed that n=O(k).Our result implies an asymptotic bound for the Ramsey number of trees. We prove that r(Tk,Tm)?k+m+o(k+m), as k+m→∞.     

On hamiltonicity of 2-connected claw-free graphs

A graph G has the hourglass property if every induced hourglass S (a tree with a degree sequence 22224) contains two non-adjacent vertices which have a common neighbor in G - V(S).For an integer k ≥ 4,...     

Vertex-disjoint quadrilaterals containing specified edges in a bipartite graph

The theory of vertex-disjoint cycles and 2-factors of graphs is the extension and generation of the well-known Hamiltonian cycles theory and it has important applications in network communication. In this paper we first prove the following result. Let G=(V ₁,V ₂;E) be a bipartite graph with |V ₁|=|V ₂|=n such that n≥2k+1, where k≥1 is an integer. If d(x)+d(y)≥?(4n+2k?1)/3? for each pair of nonadjacent vertices x and y of G with x∈V ₁ and y∈V ₂, then, for any k independent edges e ₁,…,e _k of G, G contains k vertex-disjoint quadrilaterals C ₁,…,C _k such that e _i ∈E(C _i ) for each i∈{1,…,k}. We further show that the degree condition above is sharp. If |V ₁|=|V ₂|=2k, we give degree conditions that G has a 2-factor with k vertex-disjoint quadrilaterals C ₁,…,C _k containing specified edges of G.     

The 2-extendability of 5-connected graphs on surfaces with large representativity

A graph with at least 2k+2 vertices is said to be k-extendable if any independent set of k edges in it extends to a perfect matching. We shall show that every 5-connected graph G of even order embedded on a closed surface F², except the sphere, is 2-extendable if ρ(G)?7−2χ(F²), where ρ(G) stands for the representativity of G on F² and χ(F²) for the Euler characteristic of F².