A left-first search algorithm for planar graphs

We give anO(|V(G)|)-time algorithm to assign vertical and horizontal segments to the vertices of any bipartite plane graphG so that (i) no two segments have an interior point in common, and (ii) two segments touch each other if and only if the corresponding vertices are adjacent. As a corollary, we obtain a strengthening of the following theorem of Ringel and Petrovič. The edges of any maximal bipartite plane graphG with outer facebwb′w′ can be colored by two colors such that the color classes form spanning trees ofG−b andG−b′, respectively. Furthermore, such a coloring can be found in linear time. Our method is based on a new linear-time algorithm for constructing bipolar orientations of 2-connected plane graphs. The research of H. de Fraysseix and P. O. de Mendez was supported by ESPRIT Basic Research Action No. 7141 (ALCOM II). J. Pach's research was supported by NSF Grant CCR-91-22103, OTKA-4269, and ALCOM II.     

On a maximal torus in subgroups of a general linear group

Let k be a field, K/k a finite extension of it of degree n. We denote G=Aut(kK), G_o=Aut(k K) and fix in K a basis ω₁,...,ω_n over k. In this basis, to any automorphism group of kK there corresponds a matrix group, which is denoted by the same symbol. Let G′≤G., In this paper, the conditions under which G′⊎G_o is a maximal torus in G′ are studied. The calculation of N_G′(G′⊎G_o) is carried out, provided that thee conditions are fulfilled. The case G′=SL (kK) is of particular interset. It is known that for Galois extensions and for extensions of algebraic number fields, G′⊎G_o is a maximal torus in G′. Bibligraphy: 2 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995. pp. 15–22.     

The binding number of line graphs and total graphs

D.R. Woodall [7] introduced the concept of the binding number of a graphG, bind (G), and proved that bind(G)≦(|V(G)|−1)/(|V(G)|−ρ(G)). In this paper, some properties of a graph with bind(G)=(|V(G)|−1)/(|V(G)|−ρ(G)) are given, and the binding number of some line graphs and total graphs are determined.     

Acyclic edge colorings of planar graphs and seriesparallel graphs

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G, denoted by a′(G), is the least number of colors in an acyclic edge coloring of G. Alon et al. conjectured that a′(G) ⩽ Δ(G) + 2 for any graphs. For planar graphs G with girth g(G), we prove that a′(G) ⩽ max{2Δ(G) − 2, Δ(G) + 22} if g(G) ⩾ 3, a′(G) ⩽ Δ(G) + 2 if g(G) ⩾ 5, a′(G) ⩽ Δ(G) + 1 if g(G) ⩾ 7, and a′(G) = Δ(G) if g(G) ⩾ 16 and Δ(G) ⩾ 3. For series-parallel graphs G, we have a′(G) ⩽ Δ(G) + 1. This work was supported by National Natural Science Foundation of China (Grant No. 10871119) and Natural Science Foundation of Shandong Province (Grant No. Y2008A20).     

On the total domination subdivision numbers in graphs

A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γ _t (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sd_γt (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. Karami, Khoeilar, Sheikholeslami and Khodkar, (Graphs and Combinatorics, 2009, 25, 727–733) proved that for any connected graph G of order n ≥ 3, sd_{γ t} (G) ≤ 2γ _t (G) − 1 and posed the following problem: Characterize the graphs that achieve the aforementioned upper bound. In this paper we first prove that sd_{γ t} (G) ≤ 2α′(G) for every connected graph G of order n ≥ 3 and δ(G) ≥ 2 where α′(G) is the maximum number of edges in a matching in G and then we characterize all connected graphs G with sd_{γ t} (G)=2γ _t (G)−1.     

Adjacent strong edge colorings and total colorings of regular graphs

It is conjectured that χas(G) = χt(G) for every k-regular graph G with no C5 component (k 2). This conjecture is shown to be true for many classes of graphs, including: graphs of type 1; 2-regular, 3-regular and (|V (G)| - 2)-regular graphs; bipartite graphs; balanced complete multipartite graphs; k-cubes; and joins of two matchings or cycles.     

Extensions on 2-edge connected 3-regular up-embeddable graphs

1.IntroductionLetHbea3-regulargraph.Forel,e2,e3EE(H)(el,eZande3areallowedtobethesame),weaddthreenewvenicesal)wZandw3inel,eZande3respectively.ChoosingueV(H),andthenjoiningutofi(i=1,2,3),weatlastobtaina3-regulargraphG.Or,inotherwords,wesaythatGisobtainedfromHbyanM-extension.DenoteG=M(u)(H)(seeFig.1).LetHbea3-regulargraph.Takingel,eZEE(H)(elandeZareallowedtobethesame),weputtwodistinctvenicestvian0dwZinelandeZrespectively,andaddtwodistinctvenicesu,v4V(H),thenjoinutoal,joinvtowZandjoin…     

A generalization of Dirac’s theorem for K(1,3)-free graphs

It is known that if a 2-connected graphG of sufficiently large ordern satisfies the property that the union of the neighborhoods of each pair of vertices has cardinality at leastn/2, thenG is hamiltonian. In this paper, we obtain a similar generalization of Dirac’s Theorem forK(1,3)-free graphs. In particular, we show that ifG is a 2-connectedK(1,3)-free graph of ordern with the cardinality of the union of the neighborhoods of each pair of vertices at least (n+1)/3, thenG is hamiltonian. We also investigate several other related properties inK(1,3)-free graphs such as traceability, hamiltonian-connectedness, and pancyclicity. Partially Supported by O. N. R. Contract Number N00014-88-K-0070. Partially Supported by O. N. R. Contract Number N00014-85-K-0694.     

The chromatic number of the product of two 4-chromatic graphs is 4

For any graphG and numbern≧1 two functionsf, g fromV(G) into {1, 2, ...,n} are adjacent if for all edges (a, b) ofG, f(a) ≠g(b). The graph of all such functions is the colouring graph ℒ(G) ofG. We establish first that χ(G)=n+1 implies χ(ℒ(G))=n iff χ(G ×H)=n+1 for all graphsH with χ(H)≧n+1. Then we will prove that indeed for all 4-chromatic graphsG χ(ℒ(G))=3 which establishes Hedetniemi’s [3] conjecture for 4-chromatic graphs. This research was supported by NSERC grant A7213     

New results on 3-domination critical graphs

A graph G is said to be 3-domination critical if its domination number γ(G) = 3 and γ(G + e) = 2 for any edge e not contained in G. In this paper we first establish some structural properties of 3-domination critical graphs with diameter equal to 3. In particular, this allows us to characterize a special family of 3-domination critical graphs which contains those with minimum degree one. Moreover, we show that if the minimum degree of a 3-domination critical graph G is at least 3, then α(G) ≤ κ(G) + 1 or G is superconnected, where α(G) is the independence number and κ(G) is the vertex-connectivity of G.     

On the maximal number of certain subgraphs inK r -free graphs

Given two graphsH andG, letH(G) denote the number of subgraphs ofG isomorphic toH. We prove that ifH is a bipartite graph with a one-factor, then for every triangle-free graphG withn verticesH(G) H(T ₂(n)), whereT ₂(n) denotes the complete bipartite graph ofn vertices whose colour classes are as equal as possible. We also prove that ifK is a completet-partite graph ofm vertices,r > t, n max(m, r – 1), then there exists a complete (r – 1)-partite graphG* withn vertices such thatK(G) K(G*) holds for everyK _r -free graphG withn vertices. In particular, in the class of allK _r -free graphs withn vertices the complete balanced (r – 1)-partite graphT _r–1(n) has the largest number of subgraphs isomorphic toK _t (t < r),C ₄,K _2,3. These generalize some theorems of Turán, Erdös and Sauer.Dedicated to Paul Turán on his 80th Birthday     

On the sum of the reciprocals of cycle lengths in sparse graphs

For a graphG let ℒ(G)=Σ{1/k contains a cycle of lengthk}. Erdős and Hajnal [1] introduced the real functionf(α)=inf {ℒ (G)|E(G)|/|V(G)|≧α} and suggested to study its properties. Obviouslyf(1)=0. We provef (k+1/k)≧(300k logk)⁻¹ for all sufficiently largek, showing that sparse graphs of large girth must contain many cycles of different lengths.     

Stability in CAN-free graphs

A class $\text{F}$ of graphs characterized by three forbidden subgraphs C, A, N is considered; C is the claw (the unique graph with degree sequence (1, 1, 1, 3)), A is the antenna (a graph with degree sequence (1, 2, 2, 3, 3, 3) which does not contain C), and N is the net (the unique graph with degree sequence (1, 1, 1, 3, 3, 3)). These graphs are called CAN-free. A construction is described which associates with every CAN-free graph G another CAN-free graph G′ with strictly fewer nodes than G and with stbility number α(G′) = α(G) ? 1. This gives a good algorithm for determining the stability number of CAN-free graphs.     

Expanders obtained from affine transformations

A bipartite graphG=(U, V, E) is an (n, k, δ, α) expander if |U|=|V|=n, |E|≦kn, and for anyX⊆U with |X|≦αn, |Γ _G (X)|≧(1+δ(1−|X|/n)) |X|, whereΓ _G (X) is the set of nodes inV connected to nodes inX with edges inE. We show, using relatively elementary analysis in linear algebra, that the problem of estimating the coefficientδ of a bipartite graph is reduced to that of estimating the second largest eigenvalue of a matrix related to the graph. In particular, we consider the case where the bipartite graphs are defined from affine transformations, and obtain some general results on estimating the eigenvalues of the matrix by using the discrete Fourier transform. These results are then used to estimate the expanding coefficients of bipartite graphs obtained from two-dimensional affine transformations and those obtained from one-dimensional ones.     

On the Chromatic Number of H-Free Graphs of Large Minimum Degree

The problem of determining the chromatic number of H-free graphs has been well studied, with particular attention to K _r -free graphs with large minimum degree. Recent progress has been made for triangle-free graphs on n vertices with minimum degree larger than n/3. In this paper, we determine the family of r-colorable graphs H_r{\mathcal{H}_r}, such that if H ? H_r{H \in \mathcal{H}_r}, there exists a constant C < (r − 2)/(r − 1) such that any H-free graph G on n vertices with δ(G) > Cn has chromatic number bounded above by a function dependent only on H and C. A value of C < (r − 2)/(r − 1) is given for every H ? H_r{H \in \mathcal{H}_r}, with particular attention to the case when χ(H) = 3.     

Computing the genus of the 2-amalgamations of graphs

The above authors [2] and S. Stahl [3] have shown that if a graphG is the 2-amalgamation of subgraphsG ₁ andG ₂ (namely ifG=G ₁∪G ₂ andG ₁∩G ₂={x, y}, two distinct points) then the orientable genus ofG,γ(G), is given byγ(G)=γ(G ₁)+γ(G ₂)+ε, whereε=0,1 or −1. In this paper we sharpen that result by giving a means by whichε may be computed exactly. This result is then used to give two irreducible graphs for each orientable surface.     

Large centralizers in finite solvable groups

In 1955 R. Brauer and K. A. Fowler showed that ifG is a group of even order >2, and the order |Z(G)| of the center ofG is odd, then there exists a strongly real) elementx∈G−Z whose centralizer satisfies|C _G(x)|>|G|^1/3. In Theorem 1 we show that every non-abeliansolvable groupG contains an elementx∈G−Z such that|C _G(x)|>[G:G′∩Z]^1/2 (and thus|C _G(x)|>|G|^1/3). We also note that if non-abelianG is either metabelian, nilpotent or (more generally) supersolvable, or anA-group, or any Frobenius group, then|C _G(x)|>|G|^1/2 for somex∈G−Z. In Theorem 2 we prove that every non-abelian groupG of orderp ^mqⁿ (p, q primes) contains a proper centralizer of order >|G|^1/2. Finally, in Theorem 3 we show that theaverage |C(x)|, x∈G, is ≧c|G| ^1/3 for metabelian groups, wherec is constant and the exponent 1/3 is best possible.     

On the adjacent-vertex-strongly-distinguishing total coloring of graphs

For any vertex u∈V(G), let T_N(U)={u}∪{uv|uv∈E(G), v∈v(G)}∪{v∈v(G)|uv∈E(G)}and let f be a total k-coloring of G. The total-color neighbor of a vertex u of G is the color set C_f(u)={f(x)|x∈TN(U)}. For any two adjacent vertices x and y of V(G)such that C_f(x)≠C_f(y), we refer to f as a k-avsdt-coloring of G("avsdt"is the abbreviation of"adjacent-vertex-strongly- distinguishing total"). The avsdt-coloring number of G, denoted by X_(ast)(G), is the minimal number of colors required for a avsdt-coloring of G. In this paper, the avsdt-coloring numbers on some familiar graphs are studied, such as paths, cycles, complete graphs, complete bipartite graphs and so on. We proveΔ(G) 1≤X_(ast)(G)≤Δ(G) 2 for any tree or unique cycle graph G.     

Acyclic Edge Coloring of Planar Graphs Without Small Cycles

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G, denoted by a′(G), is the least number of colors in an acyclic edge coloring of G. Alon et al. conjectured that a′(G) ≤ Δ(G) + 2 for any graphs. In this paper, it is shown that the conjecture holds for planar graphs without 4- and 5-cycles or without 4- and 6-cycles.     

Graphs whose critical groups have larger rank

The critical group C(G) of a graph G is a refinement of the number of spanning trees of the graph and is closely connected with the Laplacian matrix. Let r(G) be the minimum number of generators (i.e., the rank) of the group C(G) and β(G) be the number of independent cycles of G. In this paper, some forbidden induced subgraphs are given for r(G) = n − 3 and all graphs with r(G) = β(G) = n − 3 are characterized.