Biased partitions and judicious <Emphasis Type="Italic">k</Emphasis>-partitions of graphs

Let G = (V,E) be a graph with m edges. For reals p ∈ [0, 1] and q = 1- p, let m_p(G) be the minimum of qe(V₁) +pe(V₂) over partitions V = V₁ ∪ V₂, where e(V_i) denotes the number of edges spanned by V_i. We show that if mp(G) = pqm-δ, then there exists a bipartition V₁, V₂ of G such that e(V₁) ≤ p²m - δ + p√m/2 + o(√m) and e(V₂) ≤ q²m - δ + q √m/2 + o(√m) for δ = o(m^2/3). This is sharp for complete graphs up to the error term o(√m). For an integer k ≥ 2, let fk(G) denote the maximum number of edges in a k-partite subgraph of G. We prove that if fk(G) = (1 - 1/k)m + α, then G admits a k-partition such that each vertex class spans at most m/k² - Ω(m/k^7.5) edges for α = Ω(m/k⁶). Both of the above improve the results of Bollobás and Scott.     

On the planarity of jump graphs

For a graph G of size m1 and edge-induced subgraphs F and H of size k (1km), the subgraph H is said to be obtained from F by an edge jump if there exist four distinct vertices u,v,w, and x in G such that uvE(F), wxE(G)−E(F), and H=F−uv+wx. The minimum number of edge jumps required to transform F into H is the k-jump distance from F to H. For a graph G of size m1 and an integer k with 1km, the k-jump graph J_k(G) is that graph whose vertices correspond to the edge-induced subgraphs of size k of G and where two vertices of J_k(G) are adjacent if and only if the k-jump distance between the corresponding subgraphs is 1. All connected graphs G for which J₂(G) is planar are determined.     

A note on linearization of actions of finitely semisimple Hopf algebras on local algebras

Let H be a Hopf algebra over a field k and let H A → A, h a → h.a, be an action of H on a commutative local Noetherian kalgebra (A, m). We say that this action is linearizable if there exists a minimal system x₁, …, x_n of generators of the maximal ideal m such that h.x_i ε kx₁ + …+ kx_n for all h ε H and i = 1, …, n. In the paper we prove that the actions from a certain class are linearizable (see Theorem 4), and we indicate some consequences of this fact.     

Number of minimum vertex cuts in transitive graphs

Let G be a k-regular vertex transitive graph with connectivity κ(G)=k and let m_k(G) be the number of vertex cuts with k vertices. Define m(n,k)=min{m_k(G): GT_n,k}, where T_n,k denotes the set of all k-regular vertex transitive graphs on n vertices with κ(G)=k. In this paper, we determine the exact values of m(n,k).     

Measurements of edge-uncolorability

Cubic bridgeless graphs with chromatic index four are called uncolorable. We introduce parameters measuring the uncolorability of those graphs and relate them to each other. For k=2,3, let c_k be the maximum size of a k-colorable subgraph of a cubic graph G=(V,E). We consider r₃=|E|−c₃ and . We show that on one side r₃ and r₂ bound each other, but on the other side that the difference between them can be arbitrarily large. We also compare them to the oddness ω of G, the smallest possible number of odd circuits in a 2-factor of G. We construct cyclically 5-edge connected cubic graphs where r₃ and ω are arbitrarily far apart, and show that for each 1c<2 there is a cubic graph such that ωcr₃. For k=2,3, let ζ_k denote the largest fraction of edges that can be k-colored. We give best possible bounds for these parameters, and relate them to each other.     

Some results on characterizing the edges of connected graphs with a given domination number

A dominating set for a graph G = (V, E) is a subset of vertices V′ V such that for all v ε V − V′ there exists some u ε V′ for which {v, u} ε E. The domination number of G is the size of its smallest dominating set(s). For a given graph G with minimum size dominating set D, let m₁ (G, D) denote the number of edges that have neither endpoint in D, and let m₂ (G, D) denote the number of edges that have at least one endpoint in D. We characterize the possible values that the pair (m₁ (G, D), m₂ (G, D)) can attain for connected graphs having a given domination number.     

Zagreb indices of graphs

The first Zagreb index M₁(G) is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index M₂(G) is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying molecular graph G. In this paper, we obtain lower and upper bounds on the first Zagreb index M₁(G) of G in terms of the number of vertices (n), number of edges (m), maximum vertex degree (Δ), and minimum vertex degree (δ). Using this result, we find lower and upper bounds on M₂(G). Also, we present lower and upper bounds on M₂(G) +M₂(G) in terms of n, m, Δ, and δ, where G denotes the complement of G. Moreover, we determine the bounds on first Zagreb coindex M₁(G) and second Zagreb coindex M₂(G). Finally, we give a relation between the first Zagreb index and the second Zagreb index of graph G.     

On neighborhood condition for graphs to have [a,b]-factors

Let G be a graph of order n, and let a and b be integers such that 1a<b. Let δ(G) be the minimum degree of G. Then we prove that if δ(G)(k−1)a, n(a+b)(k(a+b)−2)/b, and |N_G(x₁)N_G(x₂)N_G(x_k)|an/(a+b) for any independent subset {x₁,x₂,…,x_k} of V(G), where k2, then G has an [a,b]-factor. This result is best possible in some sense.     

Flawless 0-sequences and Hilbert functions of Cohen-Macaulay integral domains

Let A = A₀A₁ be a commutative graded ring such that (i) A₀ = k a field, (ii) A = k[A₁] and (iii) dim_k A₁ < ∞. It is well known that the formal power series ∑^∞_{n = 0} (dim_kA_n)λ ⁿ is of the form (h₀ + h₁λ + + h_sλ^s)/(1 − λ)^dimA with each h_iε . We are interested in the sequence (h₀, h₁,…,h_s), called the h-vector of A, when A is a Cohen–Macaulay integral domain. In this paper, after summarizing fundamental results (Section 1), we study h-vectors of certain Gorenstein domains (Section 2) and find some examples of h-vectors arising from integrally closed level domains (Sections 3 and 4).     

ContributionsChromatic Ramsey numbers

Suppose G is a graph. The chromatic Ramsey number r_c(G) of G is the least integer m such that there exists a graph F of chromatic number m for which the following is true: for any 2-colouring of the edges of F there is a monochromatic subgraph isomorphic to G. Let M_n = min[r_c(G): χ(G) = n]. It was conjectured by Burr et al. (1976) that M_n = (n − 1)² + 1. This conjecture has been confirmed previously for n 4. In this paper, we shall prove that the conjecture is true for n = 5. We shall also improve the upper bounds for M₆ and M₇.     

Ramsey linear families and generalized subdivided graphs

The following results are obtained. (i) Let p, d, and k be fixed positive integers, and let G be a graph whose vertex set can be partitioned into parts V₁, V₂,…, V_a such that for each i at most d vertices in V₁ … V_i have neighbors in V_i+1 and r(K_k, V_i) p | V(G) |, where V_i denotes the subgraph of G induced by V_i. Then there exists a number c depending only on p, d, and k such that r(K_k, G)c | V(G) |. (ii) Let d be a positive integer and let G be a graph in which there is an independent set I V(G) such that each component of G−I has at most d vertices and at most two neighbors in I. Then r(G,G)c | V(G) |, where c is a number depending only on d. As a special case, r(G, G) 6 | V(G) | for a graph G in which all vertices of degree at least three are independent. The constant 6 cannot be replaced by one less than 4.     

Cycle-saturated graphs of minimum size

A graph G is called C_k-saturated if G contains no cycles of length k but does contain such a cycle after the addition of any new edge. Bounds are obtained for the minimum number of edges in C_k-saturated graphs for all k ≠ 8 or 10 and n sufficiently large. In general, it is shown that the minimum is between n + c₁n/k and n + c₂n/k for some positive constants c₁ and C₂. Our results provide an asymptotic solution to a 15-year-old problem of Bollobás.     

A special k-coloring for a connected k-chromatic graph

For each positive integer k we consider the smallest positive integer f(k) (dependent only on k) such that the following holds: Each connected graph G with chromatic number χ(G) = k can be properly vertex colored by k colors so that for each pair of vertices x_o and x_p in any color class there exist vertices x₁, x₂, …, x_p-1 of the same class with dist(x_i, x_i+1) f(k) for each i, 0 i p − 1. Thus, the graph is k-colorable with the vertices of each color class placed throughout the graph so that no subset of the class is at a distance > f(k) from the remainder of the class.

We prove that f(k) < 12k when the order of the graph is k(k − 2) + 1.     

The existence of Schröder designs with equal-sized holes

A holey Schröder design of type h₁^n₁h₂^n₂ … h_k^n_k (HSD(h₁^n₁h₂^n₂ … h_k^n_k)) is equivalent to a frame idempotent Schröder quasigroup (FISQ(h₁^n₁h₂^n₂ … h_k^n_k)) of order n with n_i missing subquasigroups (holes) of order h_i, (1 i k), which are disjoint and spanning, that is, Σ_{1 i k} n_ih_i = n. In this paper, it is shown that an HSD(hⁿ) exists if and only if h²n(n − 1) 0 (mod 4) with expceptions (h, n) ε {{(1,5),(1,9),(2,4)}} and the possible exception of (h, n) = (6,4).     

Pancyclicity mod k of claw-free graphs and K1,4-free graphs

For any natural number k, a graph G is said to be pancyclic mod k if it contains a cycle of every length modulo k. In this paper, we show that every K_1,4-free graph G with minimum degree δ(G)k+3 is pancyclic mod k and every claw-free graph G with δ(G)k+1 is pancyclic mod k, which confirms Thomassen's conjecture (J. Graph Theory 7 (1983) 261–271) for claw-free graphs.     

Neighborhood unions and cyclability of graphs

A graph G is said to be cyclable if for each orientation of G, there exists a set S of vertices such that reversing all the arcs of with one end in S results in a hamiltonian digraph. Let G be a 3-connected graph of order n36. In this paper, we show that if for any three independent vertices x₁, x₂ and x₃, |N(x₁)N(x₂)|+|N(x₂)N(x₃)|+|N(x₃)N(x₁)|2n+1, then G is cyclable.     

Solution to an extremal problem on bigraphic pairs with a <Emphasis Type="Italic">Z</Emphasis><Subscript>3</Subscript>-connected realization

Let S=(a₁,...,a_m; b₁,...,b_n), where a₁,...,a_m and b₁,...,b_n are two nonincreasing sequences of nonnegative integers. The pair S=(a₁,...,a_m; b₁,...,b_n) is said to be a bigraphic pair if there is a simple bipartite graph G=(X ∪ Y, E) such that a₁,...,a_m and b₁,...,b_n are the degrees of the vertices in X and Y, respectively. Let Z₃ be the cyclic group of order 3. Define σ(Z₃, m, n) to be the minimum integer k such that every bigraphic pair S=(a₁,...,a_m; b₁,...,b_n) with a_m, b_n ≥ 2 and σ(S)=a₁ +... + a_m ≥ k has a Z₃-connected realization. For n=m, Yin[Discrete Math., 339, 2018-2026 (2016)] recently determined the values of σ(Z₃, m, m) for m ≥ 4. In this paper, we completely determine the values of σ(Z₃, m, n) for m ≥ n ≥ 4.     

Largest signless Laplacian spectral radius of uniform supertrees with diameter and pendent edges (vertices)

Let \mathbb{S}(m; d; k) be the set of k-uniform supertrees with m edges and diameter d; and S₁(m; d; k) be the k-uniform supertree obtained from a loose path u₁; e₁; u₂; e₂,..., u_d; e_d; u_d+1 with length d by attaching m — d edges at vertex u_{\lfloor d/2\rfloor +1}: In this paper, we mainly determine S₁(m; d; k) with the largest signless Laplacian spectral radius in \mathbb{S}(m; d; k) for 3≤d≤m –1: We also determine the supertree with the second largest signless Laplacian spectral radius in \mathbb{S}(m; 3; k): Furthermore, we determine the unique k-uniform supertree with the largest signless Laplacian spectral radius among all k-uniform supertrees with n vertices and pendent edges (vertices).     

The extreme set condition of a graph

Let G be a simple graph. The size of any largest matching in G is called the matching number of G and is denoted by ν(G). Define the deficiency of G, def(G), by the equation def(G)=|V(G)|−2ν(G). A set of points X in G is called an extreme set if def(G−X)=def(G)+|X|. Let c₀(G) denote the number of the odd components of G. A set of points X in G is called a barrier if c₀(G−X)=def(G)+|X|. In this paper, we obtain the following:

(1) Let G be a simple graph containing an independent set of size i, where i2. If X is extreme in G for every independent set X of size i in G, then there exists a perfect matching in G.

(2) Let G be a connected simple graph containing an independent set of size i, where i2. Then X is extreme in G for every independent set X of size i in G if and only if G=(U,W) is a bipartite graph with |U|=|W|i, and |Γ(Y)||U|−i+m+1 for any Y U, |Y|=m (1mi−1).

(3) Let G be a connected simple graph containing an independent set of size i, where i2. Then X is a barrier in G for every independent set X of size i in G if and only if G=(U,W) is a bipartite graph with |U|=|W|=i, and |Γ(Y)|m+1 for any Y U, |Y|=m (1mi−1).