Tree decomposition of graphs

Let H be a tree on h≥2 vertices. It is shown that if G=(V, E) is a graph with \delta (G)\ge (|V|/2)+10h^4\sqrt{|V|\log|V|} , and h−1 divides |E|, then there is a decomposition of the edges of G into copies of H. This result is asymptotically the best possible for all trees with at least three vertices. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12, 237–251, 1998     

Connected subgraphs with small degree sums in 3‐connected planar graphs

It is well‐known that every planar graph has a vertex of degree at most five. Kotzig proved that every 3‐connected planar graph has an edge xy such that deg(x) + deg (y) ≤ 13. In this article, considering a similar problem for the case of three or more vertices that induce a connected subgraph, we show that, for a given positive integer t, every 3‐connected planar graph G with |V(G)| ≥ t has a connected subgraph H of order t such that Σ_x∈V(H) deg_G(x) ≤ 8t − 1. As a tool for proving this result, we consider decompositions of 3‐connected planar graphs into connected subgraphs of order at least t and at most 2t − 1. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 191–203, 1999     

Relationships between total domination,order, size,and maximum degree of graphs

A total dominating set, S, in a graph, G, has the property that every vertex in G is adjacent to a vertex in S. The total dominating number, γ_t(G) of a graph G is the size of a minimum total dominating set in G. Let G be a graph with no component of size one or two and with Δ(G) ≥ 3. In 6 , it was shown that |E(G)| ≤ Δ(G) (|V(G)|–γ_t(G)) and conjectured that |E(G)| ≤ (Δ(G)+3) (|V(G)|–γ_t(G))/2 holds. In this article, we prove that holds and that the above conjecture is false as there for every Δ exist Δ‐regular bipartite graphs G with |E(G)| ≥ (Δ+0.1 ln(Δ)) (|V(G)|–γ_t(G))/2. The last result also disproves a conjecture on domination numbers from 8 . © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 325–337, 2007     

Total weight choosability of graphs

A graph G = (V, E) is called (k, k′)‐total weight choosable if the following holds: For any total list assignment L which assigns to each vertex x a set L(x) of k real numbers, and assigns to each edge e a set L(e) of k′ real numbers, there is a mapping f: V∪E→? such that f(y)∈L(y) for any y∈V∪Eand for any two adjacent vertices x, x′, . We conjecture that every graph is (2, 2)‐total weight choosable and every graph without isolated edges is (1, 3)‐total weight choosable. It follows from results in [7] that complete graphs, complete bipartite graphs, trees other than K₂ are (1, 3)‐total weight choosable. Also a graph G obtained from an arbitrary graph H by subdividing each edge with at least three vertices is (1, 3)‐total weight choosable. This article proves that complete graphs, trees, generalized theta graphs are (2, 2)‐total weight choosable. We also prove that for any graph H, a graph G obtained from H by subdividing each edge with at least two vertices is (2, 2)‐total weight choosable as well as (1, 3)‐total weight choosable. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:198‐212, 2011     

Balanced judicious bipartitions of graphs

A bipartition of the vertex set of a graph is called balanced if the sizes of the sets in the bipartition differ by at most one. B. Bollobás and A. D. Scott, Random Struct Alg 21 (2002), 414–430 conjectured that if G is a graph with minimum degree of at least 2 then V(G) admits a balanced bipartition V₁, V₂ such that for each i, G has at most |E(G)|/3 edges with both ends in V_i. The minimum degree condition is necessary, and a result of B. Bollobás and A. D. Scott, J. Graph Theory 46 (2004), 131–143 shows that this conjecture holds for regular graphs G(i.e., when Δ(G)=δ(G)). We prove this conjecture for graphs G with \begin{eqnarray*}\Delta(G)\le\frac{7}{5}\delta(G)\end{eqnarray*}; hence, it holds for graphs ]ensuremathG with \begin{eqnarray*}\delta(G)\ge\frac{5}{7}|V(G)|\end{eqnarray*}. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 210–225, 2010     

Embedding graphs with bounded degree in sparse pseudorandom graphs

In this paper, we show the equivalence of somequasi-random properties for sparse graphs, that is, graphsG with edge densityp=|E(G)|/( ₂ ⁿ )=o(1), whereo(1)→0 asn=|V(G)|→∞. Our main result (Theorem 16) is the following embedding result. For a graphJ, writeN _J(x) for the neighborhood of the vertexx inJ, and letδ(J) andΔ(J) be the minimum and the maximum degree inJ. LetH be atriangle-free graph and setd _H=max{δ(J):J⊆H}. Moreover, putD _H=min{2d _H,Δ(H)}. LetC>1 be a fixed constant and supposep=p(n)≫n ⁻¹ D _H. We show that ifG is such that

Cyclic orthogonal double covers of 4-regular circulant graphs

An orthogonal double cover (ODC) of a graph H is a collection G={G_v:v∈V(H)} of |V(H)| subgraphs of H such that every edge of H is contained in exactly two members of G and for any two members G_u and G_v in G, |E(G_u)∩E(G_v)| is 1 if u and v are adjacent in H and it is 0 if u and v are nonadjacent in H. An ODC G of H is cyclic (CODC) if the cyclic group of order |V(H)| is a subgroup of the automorphism group of G. In this paper, we are concerned with CODCs of 4-regular circulant graphs.     

A remark on covering graphs

Every graph G may be transformed into a covering graph either by deletion of edges or by subdivision. Let _E (G) and _V (G) denote corresponding minimal numbers. We prove _E (G) = _V (G) for every graph G.     

Odd‐angulated graphs and cancelling factors in box products

Under what conditions is it true that if there is a graph homomorphism G □ H → G □ T, then there is a graph homomorphism H→ T? Let G be a connected graph of odd girth 2k + 1. We say that G is (2k + 1)‐angulated if every two vertices of G are joined by a path each of whose edges lies on some (2k + 1)‐cycle. We call G strongly (2k + 1)‐angulated if every two vertices are connected by a sequence of (2k + 1)‐cycles with consecutive cycles sharing at least one edge. We prove that if G is strongly (2k + 1)‐angulated, H is any graph, S, T are graphs with odd girth at least 2k + 1, and ?: G□ H→S□T is a graph homomorphism, then either ? maps G□{h} to S□{t_h} for all h∈V(H) where t_h∈V(T) depends on h; or ? maps G□{h} to {s_h}□ T for all h∈V(H) where s_h∈V(S) depends on h. This theorem allows us to prove several sufficient conditions for a cancelation law of a graph homomorphism between two box products with a common factor. We conclude the article with some open questions. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:221‐238, 2008     

Distinguishing Cartesian powers of graphs

The distinguishing number D(G) of a graph is the least integer d such that there is a d‐labeling of the vertices of G that is not preserved by any nontrivial automorphism of G. We show that the distinguishing number of the square and higher powers of a connected graph G ≠ K₂, K₃ with respect to the Cartesian product is 2. This result strengthens results of Albertson [Electron J Combin, 12 ( 1 ), #N17] on powers of prime graphs, and results of Klav?ar and Zhu [Eu J Combin, to appear]. More generally, we also prove that d(G □ H) = 2 if G and H are relatively prime and |H| ≤ |G| < 2^|H| ? |H|. Under additional conditions similar results hold for powers of graphs with respect to the strong and the direct product. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 250–260, 2006     

On 2‐factors of a bipartite graph

In this article, we consider the following problem: Given a bipartite graph G and a positive integer k, when does G have a 2‐factor with exactly k components? We will prove that if G = (V₁, V₂, E) is a bipartite graph with |V₁| = |V₂| = n ≥ 2k + 1 and δ (G) ≥ ⌈n/2⌉ + 1, then G contains a 2‐factor with exactly k components. We conjecture that if G = (V₁, V₂; E) is a bipartite graph such that |V₁| = |V₂| = n ≥ 2 and δ (G) ≥ ⌈n/2⌉ + 1, then, for any bipartite graph H = (U₁, U₂; F) with |U₁| ≤ n, |U₂| ≤ n and Δ (H) ≤ 2, G contains a subgraph isomorphic to H. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 101–106, 1999     

Unicyclic nonintegral sum graphs

A graph G = (V,E) is an integral sum graph if there exists a labeling S(G) ? Z such that V = S(G) and every two distinct vertices u, υ ∈ V are adjacent if and only if u + υ ∈ V. A connected graph G = (V,E) is called unicyclic if |V| = |E|. In this paper two infinite series are constructed of unicyclic graphs that are not integral sum graphs.     

Orthogonal factorizations of graphs

Let G be a graph with vertex set V(G) and edge set E(G). Let k₁, k₂,…,k_m be positive integers. It is proved in this study that every [0,k₁+…+k_m?m+1]‐graph G has a [0, k_i]₁^m‐factorization orthogonal to any given subgraph H with m edges. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 267–276, 2002     

Two new methods to obtain super vertex-magic total labelings of graphs

Let G=(V,E) be a finite non-empty graph, where V and E are the sets of vertices and edges of G, respectively, and |V|=n and |E|=e. A vertex-magic total labeling (VMTL) is a bijection λ from V∪E to the consecutive integers 1,2,…,n+e with the property that for every v∈V, , for some constant h. Such a labeling is super if λ(V)={1,2,…,n}. In this paper, two new methods to obtain super VMTLs of graphs are put forward. The first, from a graph G with some characteristics, provides a super VMTL to the graph kG graph composed by the disjoint unions of k copies of G, for a large number of values of k. The second, from a graph G₀ which admits a super VMTL; for instance, the graph kG, provides many super VMTLs for the graphs obtained from G₀ by means of the addition to it of various sets of edges.     

Partitioning 2-edge-colored graphs by monochromatic paths and cycles

We present results on partitioning the vertices of 2-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of Sárközy: the vertex set of every 2-edge-colored graph can be partitioned into at most 2α(G) monochromatic cycles, where α(G) denotes the independence number of G. Another direction, emerged recently from a conjecture of Schelp, is to consider colorings of graphs with given minimum degree. We prove that apart from o(|V (G)|) vertices, the vertex set of any 2-edge-colored graph G with minimum degree at least \(\tfrac{{(1 + \varepsilon )3|V(G)|}} {4}\) can be covered by the vertices of two vertex disjoint monochromatic cycles of distinct colors. Finally, under the assumption that \(\bar G\) does not contain a fixed bipartite graph H, we show that in every 2-edge-coloring of G, |V (G)| ? c(H) vertices can be covered by two vertex disjoint paths of different colors, where c(H) is a constant depending only on H. In particular, we prove that c(C ₄)=1, which is best possible.     

Hall parameters of complete and complete bipartite graphs

Given a graph G, for each υ ∈V(G) let L(υ) be a list assignment to G. The well‐known choice number c(G) is the least integer j such that if |L(υ)| ≥j for all υ ∈V(G), then G has a proper vertex colouring ? with ?(υ) ∈ L (υ) (?υ ∈V(G)). The Hall number h(G) is like the choice number, except that an extra non‐triviality condition, called Hall's condition, has to be satisfied by the list assignment. The edge‐analogue of the Hall number is called the Hall index, h′(G), and the total analogue is called the total Hall number, h″(G), of G. If the stock of colours from which L(υ) is selected is restricted to a set of size k, then the analogous numbers are called k‐restricted, or restricted, Hall parameters, and are denoted by h_k(G), h′_k(G) and h″_k(G). Our main object in this article is to determine, or closely bound, h′(K), h″(K_n), h′(K_m,n) and h′_k(K_m,n). We also answer some hitherto unresolved questions about Hall parameters. We show in particular that there are examples of graphs G with h′(G)?h′(G ? e)>1. We show that there are examples of graphs G and induced subgraphs H with h_k(G)<h_k(H) [this phenomenon cannot occur with unrestricted Hall numbers]. We also give an example of a graph G and an integer k such that h_k(G)<χ(G)<h(G). © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 208–237, 2002     

A lower bound on the total signed domination numbers of graphs

Let G be a finite connected simple graph with a vertex set V(G)and an edge set E(G). A total signed domination function of G is a function f:V(G)∪E(G)→{-1,1}.The weight of f is W(f)=∑_(x∈V)(G)∪E(G))f(X).For an element x∈V(G)∪E(G),we define f[x]=∑_(y∈NT[x])f(y).A total signed domination function of G is a function f:V(G)∪E(G)→{-1,1} such that f[x]≥1 for all x∈V(G)∪E(G).The total signed domination numberγ_s~*(G)of G is the minimum weight of a total signed domination function on G. In this paper,we obtain some lower bounds for the total signed domination number of a graph G and compute the exact values ofγ_s~*(G)when G is C_n and P_n.     

Neighborhood Conditions for Fractional ID-<Emphasis Type="Italic">k</Emphasis>-factor-critical Graphs

Let G be a graph and k ≥ 2 a positive integer. Let h: E(G) → [0, 1] be a function. If \(\sum\limits_{e \mathrel\backepsilon x} {h(e) = k} \) holds for each x ∈ V (G), then we call G[F_h] a fractional k-factor of G with indicator function h where F_h = {e ∈ E(G): h(e) > 0}. A graph G is fractional independent-set-deletable k-factor-critical (in short, fractional ID-k-factor-critical), if G ? I has a fractional k-factor for every independent set I of G. In this paper, we prove that if n ≥ 9k ? 14 and for any subset X ? V (G) we have

$${N_G}(X) = V(G)if|X| \geqslant \left\lfloor {\frac{{kn}}{{3k - 1}}} \right\rfloor ;or|{N_G}(X)| \geqslant \frac{{3k - 1}}{k}|X|if|X| < \left\lfloor {\frac{{kn}}{{3k - 1}}} \right\rfloor ,$$

then G is fractional ID-k-factor-critical.

     

Eigenvalues and homology of flag complexes and vector representations of graphs

The flag complex of a graph G = (V, E) is the simplicial complex X(G) on the vertex set V whose simplices are subsets of V which span complete subgraphs of G. We study relations between the first eigenvalues of successive higher Laplacians of X(G). One consequence is the following:Theorem: Let λ₂(G) denote the second smallest eigenvalue of the Laplacian of G. If \,\frac{k}{k+1}|V|$$" align="middle" border="0"> then Applications include a lower bound on the homological connectivity of the independent sets complex I(G), in terms of a new graph domination parameter Γ(G) defined via certain vector representations of G. This in turns implies Hall type theorems for systems of disjoint representatives in hypergraphs.Received: January 2004 Revised: August 2004 Accepted: August 2004     

Domination in graphs with minimum degree two

The domination number γ(G) of a graph G = (V, E) is the minimum cardinality of a subset of V such that every vertex is either in the set or is adjacent to some vertex in the set. We show that if a connected graph G has minimum degree two and is not one of seven exceptional graphs, then γ(G)γ 2/5|V|. We also characterize those connected graphs with γ(G)γ 2/5|V|.