On <Emphasis Type="Italic">g</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis></Subscript>-colorings of nearly bipartite graphs

Let G be a simple graph, let d(v) denote the degree of a vertex v and let g be a nonnegative integer function on V (G) with 0 ≤ g(v) ≤ d(v) for each vertex v ∈ V (G). A g _c -coloring of G is an edge coloring such that for each vertex v ∈ V (G) and each color c, there are at least g(v) edges colored c incident with v. The g _c -chromatic index of G, denoted by χ′g _c (G), is the maximum number of colors such that a gc-coloring of G exists. Any simple graph G has the g _c -chromatic index equal to δ _g (G) or δ _g (G) ? 1, where \({\delta _g}\left( G \right) = \mathop {\min }\limits_{v \in V\left( G \right)} \left\lfloor {d\left( v \right)/g\left( v \right)} \right\rfloor \). A graph G is nearly bipartite, if G is not bipartite, but there is a vertex u ∈ V (G) such that G ? u is a bipartite graph. We give some new sufficient conditions for a nearly bipartite graph G to have χ′g _c (G) = δ _g (G). Our results generalize some previous results due to Wang et al. in 2006 and Li and Liu in 2011.     

Some class 1 graphs on <Emphasis Type="Italic">g</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis></Subscript>-colorings

An edge-coloring of a graph G is an assignment of colors to all the edges of G. A g _c -coloring of a graph G is an edge-coloring of G such that each color appears at each vertex at least g(v) times. The maximum integer k such that G has a g _c -coloring with k colors is called the g _c -chromatic index of G and denoted by \(\chi\prime_{g_{c}}\)(G). In this paper, we extend a result on edge-covering coloring of Zhang and Liu in 2011, and give a new sufficient condition for a simple graph G to satisfy \(\chi\prime_{g_{c}}\)(G) = δ _g (G), where \(\delta_{g}\left(G\right) = min_{v\epsilon V (G)}\left\{\lfloor\frac{d\left(v\right)}{g\left(v\right)}\rfloor\right\}\).     

<Emphasis Type="Italic">r</Emphasis>-Dynamic Chromatic Number of Some Line Graphs

An r-dynamic coloring of a graph G is a proper coloring c of the vertices such that |c(N(v))| ≥ min {r, deg(v)}, for each v ∈ V (G). The r-dynamic chromatic number of a graph G is the smallest k such that G admits an r-dynamic coloring with k colors. In this paper, we obtain the r-dynamic chromatic number of the line graph of helm graphs H_n for all r between minimum and maximum degree of H_n. Moreover, our proofs are constructive, what means that we give also polynomial time algorithms for the appropriate coloring. Finally, as the first, we define an equivalent model for edge coloring.     

Graphs with small diameter determined by their <Emphasis Type="Italic">D</Emphasis>-spectra

Let G be a connected graph with vertex set V(G) = {v₁, v₂,..., v _n }. The distance matrix D(G) = (d _ij )_n×n is the matrix indexed by the vertices of G, where d _ij denotes the distance between the vertices v _i and v _j . Suppose that λ₁(D) ≥ λ₂(D) ≥... ≥ λ _n (D) are the distance spectrum of G. The graph G is said to be determined by its D-spectrum if with respect to the distance matrix D(G), any graph having the same spectrum as G is isomorphic to G. We give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs are determined by their D-spectra.     

On <Emphasis Type="Italic">k</Emphasis>- critical 2<Emphasis Type="Italic">k</Emphasis>- connected graphs

A graph G is called an (n,k)-graph if κ(G-S)=n-|S| for any S ? V(G) with |S| ≤ k, where ?(G) denotes the connectivity of G. Mader conjectured that for k ≥ 3 the graph K_2k+2?(1-factor) is the unique (2k, k)-graph. Kriesell has settled two special cases for k = 3,4. We prove the conjecture for the general case k ≥ 5.     

<Emphasis Type="Italic">s</Emphasis>-Vertex Pancyclic Index

A graph G is vertex pancyclic if for each vertex \({v \in V(G)}\) , and for each integer k with 3 ≤ k ≤ |V(G)|, G has a k-cycle C _k such that \({v \in V(C_k)}\) . Let s ≥ 0 be an integer. If the removal of at most s vertices in G results in a vertex pancyclic graph, we say G is an s-vertex pancyclic graph. Let G be a simple connected graph that is not a path, cycle or K _1,3. Let l(G) = max{m : G has a divalent path of length m that is not both of length 2 and in a K ₃}, where a divalent path in G is a path whose interval vertices have degree two in G. The s-vertex pancyclic index of G, written vp _s (G), is the least nonnegative integer m such that L ^m (G) is s-vertex pancyclic. We show that for a given integer s ≥ 0,

$vp_s(G)\le \left\{\begin{array}{l@{\quad}l}\qquad\quad\quad\,\,\,\,\,\,\, l(G)+s+1: \quad {\rm if} \,\, 0 \le s \le 4 \\ l(G)+\lceil {\rm log}_2(s-2) \rceil+4: \quad {\rm if} \,\, s \ge 5 \end{array}\right.$

And we improve the bound for essentially 3-edge-connected graphs. The lower bound and whether the upper bound is sharp are also discussed.

     

New characterization of <Emphasis Type="Italic">S</Emphasis><Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>) by its noncommuting graph

Let G be a nonabelian group, and associate the noncommuting graph ?(G) with G as follows: the vertex set of ?(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. Let S ₄(q) be the projective symplectic simple group, where q is a prime power. We prove that if G is a group with ?(G) ? ?(S ₄(q)) then G ? S ₄(q).     

Semi-lattices of bicompact <Emphasis Type="Italic">G</Emphasis>-extensions

It is shown that any semilattice of bicompact G-extensions can be realized as a semilattice for a pseudocompact phase space with an action of a discrete group. The following examples are given: lattices of bicompact G-extensions which cannot be lattices of bicompact extensions of any Tikhonov space and a pseudocompact G-space with a discrete acting group whose semilattice of bicompact G-extensions has a countable number of minimal elements.     

<Emphasis Type="Italic">T</Emphasis>-Coloring of Certain Networks

Given a graph G and a finite set T of non-negative integers containing zero, a T-coloring of G is a non-negative integer function f defined on V(G) such that \(|f(x)-f(y)|\not \in T\) whenever \((x,y)\in E(G)\). The span of T-coloring is the difference between the largest and smallest colors, and the T-span of G is the minimum span over all T-colorings f of G. The edge span of a T-coloring is the maximum value of \(|f(x)-f(y)|\) over all edges \((x,y)\in E(G)\), and the T-edge span of G is the minimum edge span over all T-colorings f of G. In this paper, we compute T-span and T-edge span of crown graph, circular ladder and mobius ladder, generalized theta graph, series-parallel graph and wrapped butterfly network.     

On <Emphasis Type="Italic">r</Emphasis>-recognition by prime graph of <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(3) where <Emphasis Type="Italic">p</Emphasis> is an odd prime

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p and p′ are joined by an edge if there is an element in G of order pp′. We denote by k(Γ(G)) the number of isomorphism classes of finite groups H satisfying Γ(G) = Γ(H). Given a natural number r, a finite group G is called r-recognizable by prime graph if k(Γ(G)) =  r. In Shen et al. (Sib. Math. J. 51(2):244–254, 2010), it is proved that if p is an odd prime, then B _p (3) is recognizable by element orders. In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(B _p (3)), where p > 3 is an odd prime, then \({G\cong B_p(3)}\) or C _p (3). Also if Γ(G) = Γ(B ₃(3)), then \({G\cong B_3(3), C_3(3), D_4(3)}\), or \({G/O_2(G)\cong {\rm Aut}(^2B_2(8))}\). As a corollary, the main result of the above paper is obtained.     

<Emphasis Type="Italic">X</Emphasis>-decomposable finite groups for <Emphasis Type="Italic">X</Emphasis>={1, <Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">m</Emphasis> + 1, <Emphasis Type="Italic">m</Emphasis> + 2}

A normal subgroup N of a finite group G is called n-decomposable in G if N is the union of n distinct G-conjugacy classes. We study the structure of nonperfect groups in which every proper nontrivial normal subgroup is m-decomposable, m+1-decomposable, or m+2-decomposable for some positive integer m. Furthermore, we give classification for the soluble case.     

On a new property of <Emphasis Type="Italic">n</Emphasis>-poised and <Emphasis Type="Italic">G</Emphasis><Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> sets

In this paper we consider n-poised planar node sets, as well as more special ones, called G C _n sets. For the latter sets each n-fundamental polynomial is a product of n linear factors as it always holds in the univariate case. A line ? is called k-node line for a node set \(\mathcal X\) if it passes through exactly k nodes. An (n + 1)-node line is called maximal line. In 1982 M. Gasca and J. I. Maeztu conjectured that every G C _n set possesses necessarily a maximal line. Till now the conjecture is confirmed to be true for n ≤ 5. It is well-known that any maximal line M of \(\mathcal X\) is used by each node in \(\mathcal X\setminus M, \)meaning that it is a factor of the fundamental polynomial. In this paper we prove, in particular, that if the Gasca-Maeztu conjecture is true then any n-node line of G C _n set \(\mathcal {X}\) is used either by exactly \(\binom {n}{2}\) nodes or by exactly \(\binom {n-1}{2}\) nodes. We prove also similar statements concerning n-node or (n ? 1)-node lines in more general n-poised sets. This is a new phenomenon in n-poised and G C _n sets. At the end we present a conjecture concerning any k-node line.     

The Center of <Emphasis Type="Italic">Dist</Emphasis> (<Emphasis Type="Italic">GL</Emphasis>(<Emphasis Type="Italic">m</Emphasis>|<Emphasis Type="Italic">n</Emphasis>)) in Positive Characteristic

The purpose of this paper is to investigate central elements in distribution algebras D i s t(G) of general linear supergroups G = G L(m|n). As an application, we compute explicitly the center of D i s t(G L(1|1)) and its image under Harish-Chandra homomorphism.     

On <Emphasis Type="Italic">G</Emphasis>-rigid surfaces

Rigid algebraic varieties form an important class of complex varieties that exhibit interesting geometric phenomena. In this paper we propose a natural extension of rigidity to complex projective varieties with a finite group action (G-varieties) and focus on the first nontrivial case, namely, on G-rigid surfaces that can be represented as desingularizations of Galois coverings of the projective plane with Galois group G. We obtain local and global G-rigidity criteria for these G-surfaces and present several series of such surfaces that are rigid with respect to the action of the deck transformation group.     

Neighbor sum distinguishing total chromatic number of <Emphasis Type="Italic">K</Emphasis><Subscript>4</Subscript>-minor free graph

A k-total coloring of a graph G is a mapping ?: V (G) ? E(G) → {1; 2,..., k} such that no two adjacent or incident elements in V (G) ? E(G) receive the same color. Let f(v) denote the sum of the color on the vertex v and the colors on all edges incident with v: We say that ? is a k-neighbor sum distinguishing total coloring of G if f(u) 6 ≠ f(v) for each edge uv ∈ E(G): Denote χ _Σ ^″ (G) the smallest value k in such a coloring of G: Pil?niak and Wo?niak conjectured that for any simple graph with maximum degree Δ(G), χ _Σ ^″ ≤ Δ(G)+3. In this paper, by using the famous Combinatorial Nullstellensatz, we prove that for K ₄-minor free graph G with Δ(G) > 5; χ _Σ ^″ = Δ(G) + 1 if G contains no two adjacent Δ-vertices, otherwise, χ _Σ ^″ (G) = Δ(G) + 2.     

Super (<Emphasis Type="Italic">a</Emphasis>, <Emphasis Type="Italic">d</Emphasis>)-edge-antimagic total labelings of complete bipartite graphs

An (a, d)-edge-antimagic total labeling of a graph G is a bijection f from V(G) ∪ E(G) onto {1, 2,…,|V(G)| + |E(G)|} with the property that the edge-weight set {f(x) + f(xy) + f(y) | xy ∈ E(G)} is equal to {a, a + d, a + 2d,...,a + (|E(G)| ? 1)d} for two integers a > 0 and d ? 0. An (a, d)-edge-antimagic total labeling is called super if the smallest possible labels appear on the vertices. In this paper, we completely settle the problem of the super (a, d)-edge-antimagic total labeling of the complete bipartite graph K_m,n and obtain the following results: the graph K_m,n has a super (a, d)-edge-antimagic total labeling if and only if either (i) m = 1, n = 1, and d ? 0, or (ii) m = 1, n ? 2 (or n = 1 and m ? 2), and d ∈ {0, 1, 2}, or (iii) m = 1, n = 2 (or n = 1 and m = 2), and d = 3, or (iv) m, n ? 2, and d = 1.     

Influence of <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-supplemented subgroups on the structure of <Emphasis Type="Italic">p</Emphasis>-modular subgroups

A subgroup K of G is M _p -supplemented in G if there exists a subgroup B of G such that G = KB and TB < G for every maximal subgroup T of K with |K: T| = p ^α. We study the structure of the chief factor of G by using M _p -supplemented subgroups and generalize the results of Monakhov and Shnyparkov by involving the relevant results about the p-modular subgroup O ^p (G) of G.     

Some Existence Theorems on All Fractional (<Emphasis Type="Italic">g</Emphasis>, <Emphasis Type="Italic">f</Emphasis>)-factors with Prescribed Properties

Let G be a graph, and g, f: V (G) → Z⁺ with g(x) ≤ f(x) for each x ∈ V (G). We say that G admits all fractional (g, f)-factors if G contains an fractional r-factor for every r: V (G) → Z⁺ with g(x) ≤ r(x) ≤ f(x) for any x ∈ V (G). Let H be a subgraph of G. We say that G has all fractional (g, f)-factors excluding H if for every r: V (G) → Z⁺ with g(x) ≤ r(x) ≤ f(x) for all x ∈ V (G), G has a fractional r-factor F _h such that E(H) ∩ E(F _h ) = θ, where h: E(G) → [0, 1] is a function. In this paper, we show a characterization for the existence of all fractional (g, f)-factors excluding H and obtain two sufficient conditions for a graph to have all fractional (g, f)-factors excluding H.     

Strongly regular locally <Emphasis Type="Italic">GQ</Emphasis>(4,<Emphasis Type="Italic">t</Emphasis>)-graphs

Amply regular with parameters (v, k, λ, μ) we call an undirected graph with v vertices in which the degrees of all vertices are equal to k, every edge belongs to λ triangles, and the intersection of the neighborhoods of every pair of vertices at distance 2 contains exactly μ vertices. An amply regular diameter 2 graph is called strongly regular. We prove the nonexistence of amply regular locally GQ(4,t)-graphs with (t,μ) = (4, 10) and (8, 30). This reduces the classification problem for strongly regular locally GQ(4,t)-graphs to studying locally GQ(4, 6)-graphs with parameters (726, 125, 28, 20).     

<Emphasis Type="Italic">M</Emphasis> <Subscript> <Emphasis Type="Italic">p</Emphasis> </Subscript>-supplemented subgroups of finite groups

A subgroup K of G is M_p-supplemented in G if there exists a subgroup B of G such that G = KB and TB < G for every maximal subgroup T of K with |K: T| = p^α. In this paper we prove the following: Let p be a prime divisor of |G| and let H be ap-nilpotent subgroup having a Sylow p-subgroup of G. Suppose that H has a subgroup D with D_p ≠ 1 and |H: D| = p_α. Then G is p-nilpotent if and only if every subgroup T of H with |T| = |D| is M_p-supplemented in G and N_G(T_p)/C_G(T_p) is a p-group.