On n-Hamiltonian line graphs

With each nonempty graph G one can associate a graph L(G), called the line graph of G, with the property that there exists a one-to-one correspondence between E(G) and V(L(G)) such that two vertices of L(G) are adjacent if and only if the corresponding edges of G are adjacent. For integers m ≥ 2, the mth iterated line graph L^m(G) of G is defined to be L(L^m-1(G)). A graph G of order p ≥ 3 is n-Hamiltonian, 0 ≤ n ≤ p ? 3, if the removal of any k vertices, 0 ≤ k ≤ n, results in a Hamiltonian graph. It is shown that if G is a connected graph with δ(G) ≥ 3, where δ(G) denotes the minimum degree of G, then L²(G) is (δ(G) ? 3)-Hamiltonian. Furthermore, if G is 2-connected and δ(G) ≥ 4, then L²(G) is (2δ(G) ? 4)-Hamiltonian. For a connected graph G which is neither a path, a cycle, nor the graph K(1, 3) and for any positive integer n, the existence of an integer k such that L^m(G) is n-Hamiltonian for every m ≥ k is exhibited. Then, for the special case n = 1, bounds on (and, in some cases, the exact value of) the smallest such integer k are determined for various classes of graphs.     

Two characterizations of interchange graphs of complete m-partite graphs

A graph G is m-partite if its points can be partitioned into m subsets V₁,…,V_m such that every line joins a point in V_i with a point in V_j, i ≠ j. A complete m-partite graph contains every line joining V_i with V_j. A complete graph K_p has every pair of its p points adjacent. The nth interchange graph I_n(G) of G is a graph whose points can be identified with the K_n+1's of G such that two points are adjacent whenever the corresponding K_n+1's have a K_n in common.Interchange graphs of complete 2-partite and 3-partite graphs have been characterized, but interchange graphs of complete m-partite graphs for m > 3 do not seem to have been investigated. The main result of this paper is two characterizations of interchange graphs of complete m-partite graphs for m ≥ 2.     

A constructive approach for the lower bounds on the Ramsey numbers R (s,t)

Graph G is a (k, p)‐graph if G does not contain a complete graph on k vertices K_k, nor an independent set of order p. Given a (k, p)‐graph G and a (k, q)‐graph H, such that G and H contain an induced subgraph isomorphic to some K_k?1‐free graph M, we construct a (k, p + q ? 1)‐graph on n(G) + n(H) + n(M) vertices. This implies that R (k, p + q ? 1) ≥ R (k, p) + R (k, q) + n(M) ? 1, where R (s, t) is the classical two‐color Ramsey number. By applying this construction, and some its generalizations, we improve on 22 lower bounds for R (s, t), for various specific values of s and t. In particular, we obtain the following new lower bounds: R (4, 15) ≥ 153, R (6, 7) ≥ 111, R (6, 11) ≥ 253, R (7, 12) ≥ 416, and R (8, 13) ≥ 635. Most of the results did not require any use of computer algorithms. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 231–239, 2004     

Spanning 3-ended trees in k-connected K 1,4-free graphs

A tree with at most m leaves is called an m-ended tree.Kyaw proved that every connected K1,4-free graph withσ4(G)n-1 contains a spanning 3-ended tree.In this paper we obtain a result for k-connected K1,4-free graphs with k 2.Let G be a k-connected K1,4-free graph of order n with k 2.Ifσk+3(G)n+2k-2,then G contains a spanning 3-ended tree.     

On the ordering of graphs with respect to their matching numbers

The number of k-matchings in a graph G is denoted by m(G,k). If for two graphs G₁ and G₂, m(G₁, k) ≥m(G₂,k) for all k, then we write G₁ > G₂. Six classes of graphs are ordered with respect to the relation >.     

Multicoloring and Mycielski construction

The generalized Mycielskians of graphs (also known as cones over graphs) are the natural generalization of the Mycielskians of graphs (which were first introduced by Mycielski in 1955). Given a graph G and any integer p?0, one can transform G into a new graph μ_p(G), the p-Mycielskian of G. In this paper, we study the kth chromatic numbers χ_k of Mycielskians and generalized Mycielskians of graphs. We show that χ_k(G)+1?χ_k(μ(G))?χ_k(G)+k, where both upper and lower bounds are attainable. We then investigate the kth chromatic number of Mycielskians of cycles and determine the kth chromatic number of p-Mycielskian of a complete graph K_n for any integers k?1, p?0 and n?2. Finally, we prove that if a graph G is a/b-colorable then the p-Mycielskian of G, μ_p(G), is (at+b^p+1)/bt-colorable, where . And thus obtain graphs G with m(G) grows exponentially with the order of G, where m(G) is the minimal denominator of a a/b-coloring of G with χ_f(G)=a/b.     

Iteratedk-line graphs

For integersk≥2, thek-line graph L_k(G) of a graph G is defined as a graph whose vertices correspond to the complete subgraphs onk vertices in G with two distinct vertices adjacent if the corresponding complete subgraphs have 1 common vertices inG. We define iteratedk-line graphs byL _k ⁿ (G) ?L _k (L _k ^n?1 (G), whereL _k ⁰ (G) ?G. In this paper the iterated behavior of thek-line graph operator is investigated. It turns out that the behavior is quite different fork = 2 (the well-known line graph case),k = 3, and k≥4.     

Sharp bounds for decompositions of graphs into complete r-partite subgraphs

If G is a graph on n vertices and r ≥ 2, we let m_r(G) denote the minimum number of complete multipartite subgraphs, with r or fewer parts, needed to partition the edge set, E(G). In determining m_r(G), we may assume that no two vertices of G have the same neighbor set. For such reducedgraphs G, we prove that m_r(G) ≥ log₂ (n + r − 1)/r. Furthermore, for each k ≥ 0 and r ≥ 2, there is a unique reduced graph G = G(r, k) with m_r(G) = k for which equality holds. We conclude with a short proof of the known eigenvalue bound m_r(G) ≥ max{n₊ (G, n₋(G)/(r − 1)}, and show that equality holds if G = G(r, k). © 1996 John Wiley & Sons, Inc.     

A note on constructive methods for ramsey numbers

Let r(k) denote the least integer n-such that for any graph G on n vertices either G or its complement G contains a complete graph K_k on k vertices. in this paper, we prove the following lower bound for the Ramsey number r(k) by explicit construction: r(k) ≥ exp (c(Log k)^4/3[(log log k)^1/3] for some constant c> 0.     

On the chromaticity of complete multipartite graphs with certain edges added

Let P(G,λ) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P(H,λ)=P(G,λ) implies H is isomorphic to G. For integers k≥0, t≥2, denote by K((t−1)×p,p+k) the complete t-partite graph that has t−1 partite sets of size p and one partite set of size p+k. Let K(s,t,p,k) be the set of graphs obtained from K((t−1)×p,p+k) by adding a set S of s edges to the partite set of size p+k such that 〈S〉 is bipartite. If s=1, denote the only graph in K(s,t,p,k) by K⁺((t−1)×p,p+k). In this paper, we shall prove that for k=0,1 and p+k≥s+2, each graph G∈K(s,t,p,k) is chromatically unique if and only if 〈S〉 is a chromatically unique graph that has no cut-vertex. As a direct consequence, the graph K⁺((t−1)×p,p+k) is chromatically unique for k=0,1 and p+k≥3.     

Note on group distance magic complete bipartite graphs

A Γ-distance magic labeling of a graph G = (V, E) with |V| = n is a bijection ? from V to an Abelian group Γ of order n such that the weight $w(x) = \sum\nolimits_{y \in N_G (x)} {\ell (y)}$ of every vertex x ∈ V is equal to the same element µ ∈ Γ, called the magic constant. A graph G is called a group distance magic graph if there exists a Γ-distance magic labeling for every Abelian group Γ of order |V(G)|. In this paper we give necessary and sufficient conditions for complete k-partite graphs of odd order p to be ? _p -distance magic. Moreover we show that if p ≡ 2 (mod 4) and k is even, then there does not exist a group Γ of order p such that there exists a Γ-distance labeling for a k-partite complete graph of order p. We also prove that K _m,n is a group distance magic graph if and only if n + m ? 2 (mod 4).     

Note on a pursuit game played on graphs

In [1] Aigner and Fromme considered a game played on a finite graph G where m pursuers try to catch one evader. They introduced c(G) as the minimal number m of pursuers that are sufficient to catch the evader and, among other things, they asked if it is true that c(G) ≤ k whenever the maximal degree of G is at most k. In the present note we give a negative answer to this question by showing that, for all positive integers k, n (k ≥ 3), there exists a k-regular graph G with c(G) ≥ n.     

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.     

Some remarks on the odd hadwiger’s conjecture

We say that H has an odd complete minor of order at least l if there are l vertex disjoint trees in H such that every two of them are joined by an edge, and in addition, all the vertices of trees are two-colored in such a way that the edges within the trees are bichromatic, but the edges between trees are monochromatic. Gerards and Seymour conjectured that if a graph has no odd complete minor of order l, then it is (l ? 1)-colorable. This is substantially stronger than the well-known conjecture of Hadwiger. Recently, Geelen et al. proved that there exists a constant c such that any graph with no odd K _k -minor is ck√logk-colorable. However, it is not known if there exists an absolute constant c such that any graph with no odd K _k -minor is ck-colorable. Motivated by these facts, in this paper, we shall first prove that, for any k, there exists a constant f(k) such that every (496k + 13)-connected graph with at least f(k) vertices has either an odd complete minor of size at least k or a vertex set X of order at most 8k such that G–X is bipartite. Since any bipartite graph does not contain an odd complete minor of size at least three, the second condition is necessary. This is an analogous result of Böhme et al. We also prove that every graph G on n vertices has an odd complete minor of size at least n/2α(G) ? 1, where α(G) denotes the independence number of G. This is an analogous result of Duchet and Meyniel. We obtain a better result for the case α(G)= 3.     

Vertex-disjoint K_1+(K_1 ∪ K_2) in K_(1,4)-free Graphs with Minimum Degree at Least Four

A graph is said to be K1,4-free if it does not contain an induced subgraph isomorphic to K1,4.Let k be an integer with k≥2.We prove that ifG is a K1,4-free graph of order at least 11k-10 with minimum degree at least four,then G contains k vertex-disjoint copies of K1+(K1∪K2).     

Complete subgraphs with large degree sums

Let t(n, k) denote the Turán number—the maximum number of edges in a graph on n vertices that does not contain a complete graph K_k+1. It is shown that if G is a graph on n vertices with n ≥ k²(k – 1)/4 and m < t(n, k) edges, then G contains a complete subgraph K_k such that the sum of the degrees of the vertices is at least 2km/n. This result is sharp in an asymptotic sense in that the sum of the degrees of the vertices of K_k is not in general larger, and if the number of edges in G is at most t(n, k) – ? (for an appropriate ?), then the conclusion is not in general true. © 1992 John Wiley & Sons, Inc.     

Non-traceability of large connected claw-free graphs

Let G be a connected claw-free graph on n vertices. Let ς₃(G) be the minimum degree sum among triples of independent vertices in G. It is proved that if ς₃(G) ≥ n − 3 then G is traceable or else G is one of graphs G_n each of which comprises three disjoint nontrivial complete graphs joined together by three additional edges which induce a triangle K₃. Moreover, it is shown that for any integer k ≥ 4 there exists a positive integer ν(k) such that if ς₃(G) ≥ n − k, n > ν(k) and G is non-traceable, then G is a factor of a graph G_n. Consequently, the problem HAMILTONIAN PATH restricted to claw-free graphs G = (V, E) (which is known to be NP-complete) has linear time complexity O(|E|) provided that ς₃(G) ≥ . This contrasts sharply with known results on NP-completeness among dense graphs. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 75–86, 1998     

A constructive characterization of trees with at least k disjoint maximum matchings

Let H = F(v) ⊕ G(w) denote the graph obtained from F and G by identifying vertices v of F and w of G; H will be said to be obtained by surgery on F and G. A matching of a graph is a collection of edges, no two of which are incident with the same vertex. This paper presents a constructive characterization of the set S_k (k ≥ 2) of trees which have at least k disjoint maximum matchings. There are three types of surgery such that, for each k ≥ 2, S_k is the set of all trees obtainable from a star K_1.n (n ≥ k) by a finite sequence of the specified surgical operations. A constructive characterization is also given for trees with two disjoint maximum indepent vertex sets.     

On extremal k-outconnected graphs

Let G be a minimally k-connected graph with n nodes and m edges. Mader proved that if n?3k-2 then m?k(n-k), and for n?3k-1 an equality is possible if, and only if, G is the complete bipartite graph K_k,n-k. Cai proved that if n?3k-2 then m?⌊(n+k)²/8⌋, and listed the cases when this bound is tight.In this paper we prove a more general theorem, which implies similar results for minimally k-outconnected graphs; a graph is called k-outconnected from r if it contains k internally disjoint paths from r to every other node.     

On the toroidal thickness of graphs

The toroidal thickness t₁(G) of a graph G is the minimum value of k such that G is the union of k graphs each of which is embeddable on a torus. We find t₁(G_m), where G_m is the graph obtained from the complete graph K_m by removing a Hamiltonian cycle, and we show that t₁(K_n(3)) = [1/2n] for many values of n. The method of approach involves the construction of sets of triples related to Skolem triples.