Improvement on Brooks' chromatic bound for a class of graphs

Brooks' Theorem says that if for a graph G,Δ(G)=n, then G is n-colourable, unless (1) n=2 and G has an odd cycle as a component, or (2) n>2 and K_n+1 is a component of G. In this paper we prove that if a graph G has none of some three graphs (K_1,3;K₅?e and H) as an induced subgraph and if Δ(G)?6 and d(G)<Δ(G), then χ(G)<Δ(G). Also we give examples to show that the hypothesis Δ(G)?6 can not be non-trivially relaxed and the graph K₅?e can not be removed from the hypothesis. Moreover, for a graph G with none of K_1,3;K₅?e and H as an induced subgraph, we verify Borodin and Kostochka's conjecture that if for a graph G,Δ(G)?9 and d(G)<Δ(G), then χ(G)<Δ(G).     

On a co-induction question of Kechris

This note answers a question of Kechris: if H < G is a normal subgroup of a countable group G, H has property MD and G/H is amenable and residually finite, then G also has property MD. Under the same hypothesis we prove that for any action a of G, if b is a free action of G/H, and b _G is the induced action of G, then CInd _H ^G (a|H) × b _G weakly contains a. Moreover, if H < G is any subgroup of a countable group G, and the action of G on G/H is amenable, then CInd _H ^G (a|H) weakly contains a whenever a is a Gaussian action.     

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.     

Planar graphs without 5-cycles or without 6-cycles

Let G be a planar graph without 5-cycles or without 6-cycles. In this paper, we prove that if G is connected and δ(G)≥2, then there exists an edge xy∈E(G) such that d(x)+d(y)≤9, or there is a 2-alternating cycle. By using the above result, we obtain that (1) its linear 2-arboricity , (2) its list total chromatic number is Δ(G)+1 if Δ(G)≥8, and (3) its list edge chromatic number is Δ(G) if Δ(G)≥8.     

Convolution on L p -spaces of a locally compact group

We have recently shown that, for 2 < p < ∞, a locally compact group G is compact if and only if the convolution multiplication f * g exists for all f, g ∈ L ^p (G). Here, we study the existence of f * g for all f, g ∈ L ^p (G) in the case where 0 < p ≤ 2. Also, for 0 < p < ∞, we offer some necessary and sufficient conditions for L ^p (G) * L ^p (G) to be contained in certain function spaces on G.     

Edge-choosability of planar graphs without adjacent triangles or without 7-cycles

A graph G is edge-L-colorable, if for a given edge assignment L={L(e):e∈E(G)}, there exists a proper edge-coloring ? of G such that ?(e)∈L(e) for all e∈E(G). If G is edge-L-colorable for every edge assignment L with |L(e)|≥k for e∈E(G), then G is said to be edge-k-choosable. In this paper, we prove that if G is a planar graph with maximum degree Δ(G)≠5 and without adjacent 3-cycles, or with maximum degree Δ(G)≠5,6 and without 7-cycles, then G is edge-(Δ(G)+1)-choosable.     

Injective colorings of sparse graphs

Let denote the maximum average degree (over all subgraphs) of G and let χ_i(G) denote the injective chromatic number of G. We prove that if , then χ_i(G)≤Δ(G)+1; and if , then χ_i(G)=Δ(G). Suppose that G is a planar graph with girth g(G) and Δ(G)≥4. We prove that if g(G)≥9, then χ_i(G)≤Δ(G)+1; similarly, if g(G)≥13, then χ_i(G)=Δ(G).     

On the definability of verbal subgroups

We show that if G is a group of finite Morley rank, then the verbal subgroup <w(G)> is of finite width, where w is a concise word. As a byproduct, we show that if G is any abelian-by-finite group, then G ⁿ =<x ⁿ (G)> is definable. Received: 15 March 1996 / Published online: 18 July 2001     

Characterization of G 2(q), where 2 < q ≡ −1(mod 3), by order components

We prove that the simple group G ₂(q), where 2 < q ≡ ?1(mod 3), is recognizable by the set of its order components. In other words, we prove that if G is a finite group with OC(G) = OC(G ₂(q)), then G ≌ G ₂(q).     

Hamiltonian connectedness in 3-connected line graphs

We investigate graphs G such that the line graph L(G) is hamiltonian connected if and only if L(G) is 3-connected, and prove that if each 3-edge-cut contains an edge lying in a short cycle of G, then L(G) has the above mentioned property. Our result extends Kriesell’s recent result in [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306-315] that every 4-connected line graph of a claw free graph is hamiltonian connected. Another application of our main result shows that if L(G) does not have an hourglass (a graph isomorphic to K₅−E(C₄), where C₄ is an cycle of length 4 in K₅) as an induced subgraph, and if every 3-cut of L(G) is not independent, then L(G) is hamiltonian connected if and only if κ(L(G))≥3, which extends a recent result by Kriesell [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306-315] that every 4-connected hourglass free line graph is hamiltonian connected.     

Topology of random clique complexes

In a seminal paper, Erd?s and Rényi identified a sharp threshold for connectivity of the random graph G(n,p). In particular, they showed that if p?logn/n then G(n,p) is almost always connected, and if p?logn/n then G(n,p) is almost always disconnected, as n→∞.The clique complexX(H) of a graph H is the simplicial complex with all complete subgraphs of H as its faces. In contrast to the zeroth homology group of X(H), which measures the number of connected components of H, the higher dimensional homology groups of X(H) do not correspond to monotone graph properties. There are nevertheless higher dimensional analogues of the Erd?s-Rényi Theorem.We study here the higher homology groups of X(G(n,p)). For k>0 we show the following. If p=n^α, with α<−1/k or α>−1/(2k+1), then the kth homology group of X(G(n,p)) is almost always vanishing, and if −1/k<α<−1/(k+1), then it is almost always nonvanishing.We also give estimates for the expected rank of homology, and exhibit explicit nontrivial classes in the nonvanishing regime. These estimates suggest that almost all d-dimensional clique complexes have only one nonvanishing dimension of homology, and we cannot rule out the possibility that they are homotopy equivalent to wedges of a spheres.     

Chromatic coloring with a maximum color class

Let G be any graph, and also let Δ(G), χ(G) and α(G) denote the maximum degree, the chromatic number and the independence number of G, respectively. A chromatic coloring of G is a proper coloring of G using χ(G) colors. A color class in a proper coloring of G is maximum if it has size α(G). In this paper, we prove that if a graph G (not necessarily connected) satisfies χ(G)≥Δ(G), then there exists a chromatic coloring of G in which some color class is maximum. This cannot be guaranteed if χ(G)<Δ(G). We shall also give some other extensions.     

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.     

Small, nm-stable compact G-groups

We prove that if (H, G) is a small, nm-stable compact G-group, then H is nilpotent-by-finite, and if additionally NM(H) < ω or NM(H) = ω ^α for some ordinal α, then H is abelian-by-finite. Both results are significant steps towards the proof of the conjecture that each small, nm-stable compact G-group is abelian-by-finite. We provide counter-examples to the NM-gap conjecture, that is we give examples of small, nm-stable compact G-groups of infinite ordinal NM-rank.     

An upper bound on the domination number of a graph with minimum degree 2

A set S of vertices in a graph G is a dominating set of G if every vertex of V(G)?S is adjacent to some vertex in S. The minimum cardinality of a dominating set of G is the domination number of G, denoted as γ(G). Let P_n and C_n denote a path and a cycle, respectively, on n vertices. Let k₁(F) and k₂(F) denote the number of components of a graph F that are isomorphic to a graph in the family {P₃,P₄,P₅,C₅} and {P₁,P₂}, respectively. Let L be the set of vertices of G of degree more than 2, and let G−L be the graph obtained from G by deleting the vertices in L and all edges incident with L. McCuaig and Shepherd [W. McCuaig, B. Shepherd, Domination in graphs with minimum degree two, J. Graph Theory 13 (1989) 749-762] showed that if G is a connected graph of order n≥8 with δ(G)≥2, then γ(G)≤2n/5, while Reed [B.A. Reed, Paths, stars and the number three, Combin. Probab. Comput. 5 (1996) 277-295] showed that if G is a graph of order n with δ(G)≥3, then γ(G)≤3n/8. As an application of Reed’s result, we show that if G is a graph of order n≥14 with δ(G)≥2, then .     

Connectivity of iterated line graphs

Let k≥0 be an integer and L^k(G) be the kth iterated line graph of a graph G. Niepel and Knor proved that if G is a 4-connected graph, then κ(L²(G))≥4δ(G)−6. We show that the connectivity of G can be relaxed. In fact, we prove in this note that if G is an essentially 4-edge-connected and 3-connected graph, then κ(L²(G))≥4δ(G)−6. Similar bounds are obtained for essentially 4-edge-connected and 2-connected (1-connected) graphs.     

5-Shredders in 5-connected graphs

For a graph G, a subset S of V(G) is called a shredder if G−S consists of three or more components. We show that if G is a 5-connected graph with |V(G)|≥135, then the number of shredders of cardinality 5 of G is less than or equal to (2|V(G)|−10)/3.     

Translation-invariant linear forms on C0(G), C(G), Lp(G) for noncompact groups

Suppose G is a locally compact noncompact group. For abelian such G's, it is shown in this paper that L¹(G), C(G), and L^∞(G) always have discontinuous translation-invariant linear forms(TILF's) while C₀(G) and L^p(G) for 1 < p < ∞ have such forms if and only if $\text{G}\text{H}$ is a torsion group for some open σ-compact subgroup H of $\text{G}$. For σ-compact amenable G's, all the above spaces have discontinuous left TILF's.     

Minimally 3-restricted edge connected graphs

For a connected graph G=(V,E), an edge set S⊂E is a 3-restricted edge cut if G−S is disconnected and every component of G−S has order at least three. The cardinality of a minimum 3-restricted edge cut of G is the 3-restricted edge connectivity of G, denoted by λ₃(G). A graph G is called minimally 3-restricted edge connected if λ₃(G−e)<λ₃(G) for each edge e∈E. A graph G is λ₃-optimal if λ₃(G)=ξ₃(G), where , ω(U) is the number of edges between U and V?U, and G[U] is the subgraph of G induced by vertex set U. We show in this paper that a minimally 3-restricted edge connected graph is always λ₃-optimal except the 3-cube.     

The similarity problem for Fourier algebras and corepresentations of group von Neumann algebras

Let G be a locally compact group, and let A(G) and VN(G) be its Fourier algebra and group von Neumann algebra, respectively. In this paper we consider the similarity problem for A(G): Is every bounded representation of A(G) on a Hilbert space H similar to a *-representation? We show that the similarity problem for A(G) has a negative answer if and only if there is a bounded representation of A(G) which is not completely bounded. For groups with small invariant neighborhoods (i.e. SIN groups) we show that a representation π:A(G)→B(H) is similar to a *-representation if and only if it is completely bounded. This, in particular, implies that corepresentations of VN(G) associated to non-degenerate completely bounded representations of A(G) are similar to unitary corepresentations. We also show that if G is a SIN, maximally almost periodic, or totally disconnected group, then a representation of A(G) is a *-representation if and only if it is a complete contraction. These results partially answer questions posed in Effros and Ruan (2003) [7] and Spronk (2002) [25].