Average distance,minimum degree,and spanning trees

The average distance μ(G) of a connected graph G of order n is the average of the distances between all pairs of vertices of G, i.e., μ(G) = ()⁻¹ Σ_{x,y}⊂V(G) d_G(x, y), where V(G) denotes the vertex set of G and d_G(x, y) is the distance between x and y. We prove that every connected graph of order n and minimum degree δ has a spanning tree T with average distance at most . We give improved bounds for K₃‐free graphs, C₄‐free graphs, and for graphs of given girth. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 1–13, 2000     

The circular chromatic number of the Mycielskian of G

In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph μ(G), we now call the Mycielskian of G, which has the same clique number as G and whose chromatic number equals χ(G) + 1. Chang, Huang, and Zhu [G. J. Chang, L. Huang, & X. Zhu, Discrete Math, to appear] have investigated circular chromatic numbers of Mycielskians for several classes of graphs. In this article, we study circular chromatic numbers of Mycielskians for another class of graphs G. The main result is that χ_c(μ(G)) = χ(μ(G)), which settles a problem raised in [G. J. Chang, L. Huang, & X. Zhu, Discrete Math, to appear, and X. Zhu, to appear]. As χ_c(G) = and χ(G) = , consequently, there exist graphs G such that χ_c(G) is as close to χ(G) − 1 as you want, but χ_c(μ(G)) = χ(μ(G)). © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 63–71, 1999     

Subgraphs smaller than the girth

For graphs A, B, let () denote the number of subsets of nodes of A for which the induced subgraph is B. If G and H both have girth > k, and if () = () for every k-node tree T, then for every k-node forest F, () = (). Say the spread of a tree is the number of nodes in a longest path. If G is regular of degree d, on n nodes, with girth > k, and if F is a forest of total spread ≤k, then the value of () depends only on n and d.     

On local connectivity of graphs with given clique number

For a vertex v of a graph G, we denote by d(v) the degree of v. The local connectivity κ(u, v) of two vertices u and v in a graph G is the maximum number of internally disjoint u –v paths in G, and the connectivity of G is defined as κ(G)=min{κ(u, v)|u, v∈V(G)}. Clearly, κ(u, v)?min{d(u), d(v)} for all pairs u and v of vertices in G. Let δ(G) be the minimum degree of G. We call a graph G maximally connected when κ(G)=δ(G) and maximally local connected when for all pairs u and v of distinct vertices in G. In 2006, Hellwig and Volkmann (J Graph Theory 52 (2006), 7–14) proved that a connected graph G with given clique number ω(G)?p of order n(G) is maximally connected when As an extension of this result, we will show in this work that these conditions even guarantee that G is maximally local connected. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 192–197, 2010     

The chromatic index of graphs of even order with many edges

We show that, for r = 1, 2, a graph G with 2n + 2 (≥6) vertices and maximum degree 2n + 1 - r is of Class 2 if and only if |E(G/v)| > () - rn, where v is a vertex of G of minimum degree, and we make a conjecture for 1 ≤ r ≤ n, of which this result is a special case. For r = 1 this result is due to Plantholt.     

Semilinear wave equation with time dependent potential

We consider the following semilinear wave equation: (1) for (t,x) ∈ ?_t × ?. We prove that if the potential V(t,x) is a measurable function that satisfies the following decay assumption: ∣V(t,x)∣?C(1+t)(1+∣x∣) for a.e. (t,x) ∈ ?_t × ? where C, σ₀>0 are real constants, then for any real number λ that satisfies there exists a real number ρ(f,g,λ)>0 such that the equation has a global solution provided that 0<ρ?ρ(f,g,λ). Copyright © 2004 John Wiley & Sons, Ltd.     

Properly colored subgraphs and rainbow subgraphs in edge‐colorings with local constraints

We consider a canonical Ramsey type problem. An edge‐coloring of a graph is called m‐good if each color appears at most m times at each vertex. Fixing a graph G and a positive integer m, let f(m, G) denote the smallest n such that every m‐good edge‐coloring of K_n yields a properly edge‐colored copy of G, and let g(m, G) denote the smallest n such that every m‐good edge‐coloring of K_n yields a rainbow copy of G. We give bounds on f(m, G) and g(m, G). For complete graphs G = K_t, we have c₁mt²/ln t ≤ f(m, K_t) ≤ c₂mt², and cmt³/ln t ≤ g(m, K_t) ≤ cmt³/ln t, where c₁, c₂, c, c are absolute constants. We also give bounds on f(m, G) and g(m, G) for general graphs G in terms of degrees in G. In particular, we show that for fixed m and d, and all sufficiently large n compared to m and d, f(m, G) = n for all graphs G with n vertices and maximum degree at most d. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2003     

Nonstationary Stokes system in weighted Sobolev spaces

We consider an initial‐boundary value problem for nonstationary Stokes system in a bounded domain Omega??³ with slip boundary conditions. We assume that Ω is crossed by an axis L. Let us introduce the following weighted Sobolev spaces with finite norms: and where ?(x) = dist{x, L}. We proved the result. Given the external force f∈L_{2, ?µ}(Ω^T), initial velocity v₀∈H(Ω), µ∈?₊\? there exist velocity v∈H(Ω^T) and the pressure p, ?p∈L_{2, ?µ}(Ω^T) and a constant c, independent of v, p, f, such that As we consider the Stokes system in weighted Sobolev spaces the following two things must be used:

• 1. the slip boundary condition and
• 2. the Helmholtz–Weyl decomposition.

Copyright © 2010 John Wiley & Sons, Ltd.     

Doubly transitive 2‐factorizations

Let be a 2‐factorization of the complete graph K_v admitting an automorphism group G acting doubly transitively on the set of vertices. The vertex‐set V(K_v) can then be identified with the point‐set of AG(n, p) and each 2‐factor of is the union of p‐cycles which are obtained from a parallel class of lines of AG(n, p) in a suitable manner, the group G being a subgroup of A G L(n, p) in this case. The proof relies on the classification of 2‐(v, k, 1) designs admitting a doubly transitive automorphism group. The same conclusion holds even if G is only assumed to act doubly homogeneously. © 2006 Wiley Periodicals, Inc. J Combin Designs     

Upper total domination in claw‐free graphs

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S (other than itself). The maximum cardinality of a minimal total dominating set of G is the upper total domination number of G, denoted by Γ_t(G). We establish bounds on Γ_t(G) for claw‐free graphs G in terms of the number n of vertices and the minimum degree δ of G. We show that if if , and if δ ≥ 5. The extremal graphs are characterized. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 148–158, 2003     

Explicit sparse almost‐universal graphs for ${\bf {{\cal G}(n, {k \over n})}}$

For any integer n, let be a probability distribution on the family of graphs on n vertices (where every such graph has nonzero probability associated with it). A graph Γ is ‐almost‐universal if Γ satisifies the following: If G is chosen according to the probability distribution , then G is isomorphic to a subgraph of Γ with probability 1 ‐ . For any p ∈ [0,1], let (n,p) denote the probability distribution on the family of graphs on n vertices, where two vertices u and v form an edge with probability p, and the events {u and v form an edge}; u,v ∈ V (G) are mutually independent. For k ≥ 4 and n sufficiently large we construct a ‐almost‐universal‐graph on n vertices and with O(n)polylog(n) edges, where q = ? ? for such k ≤ 6, and where q = ? ? for k ≥ 7. The number of edges is close to the lower bound of Ω( ) for the number of edges in a universal graph for the family of graphs with n vertices and maximum degree k. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Circular choosability via combinatorial Nullstellensatz

A p‐list assignment L of a graph G assigns to each vertex v of G a set of permissible colors. We say G is L‐(P, q)‐colorable if G has a (P, q)‐coloring h such that h(v) ? L(v) for each vertex v. The circular list chromatic number of a graph G is the infimum of those real numbers t for which the following holds: For any P, q, for any P‐list assignment L with , G is L‐(P, q)‐colorable. We prove that if G has an orientation D which has no odd directed cycles, and L is a P‐list assignment of G such that for each vertex v, , then G is L‐(P, q)‐colorable. This implies that if G is a bipartite graph, then , where is the maximum average degree of a subgraph of G. We further prove that if G is a connected bipartite graph which is not a tree, then . © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 190–204, 2008     

Balloons,cut‐edges,matchings, and total domination in regular graphs of odd degree

A balloon in a graph G is a maximal 2‐edge‐connected subgraph incident to exactly one cut‐edge of G. Let b(G) be the number of balloons, let c(G) be the number of cut‐edges, and let α′(G) be the maximum size of a matching. Let ${\mathcal{F}}_{{{n}},{{r}}}A balloon in a graph G is a maximal 2‐edge‐connected subgraph incident to exactly one cut‐edge of G. Let b(G) be the number of balloons, let c(G) be the number of cut‐edges, and let α′(G) be the maximum size of a matching. Let ${\mathcal{F}}_{{{n}},{{r}}}$ be the family of connected (2r+1)‐regular graphs with n vertices, and let ${{b}}={{max}}\{{{b}}({{G}}): {{G}}\in {\mathcal{F}}_{{{n}},{{r}}}\}$. For ${{G}}\in{\mathcal{F}}_{{{n}},{{r}}}$, we prove the sharp inequalities c(G)?[r(n?2)?2]/(2r²+2r?1)?1 and α′(G)?n/2?rb/(2r+1). Using b?[(2r?1)n+2]/(4r²+4r?2), we obtain a simple proof of the bound proved by Henning and Yeo. For each of these bounds and each r, the approach using balloons allows us to determine the infinite family where equality holds. For the total domination number γ_t(G) of a cubic graph, we prove γ_t(G)?n/2?b(G)/2 (except that γ_t(G) may be n/2?1 when b(G)=3 and the balloons cover all but one vertex). With α′(G)?n/2?b(G)/3 for cubic graphs, this improves the known inequality γ_t(G)?α′(G). © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 116–131, 2010     

Acyclic edge chromatic number of outerplanar 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 χ(G), is the least number of colors in an acyclic edge coloring of G. In this paper, we determine completely the acyclic edge chromatic number of outerplanar graphs. The proof is constructive and supplies a polynomial time algorithm to acyclically color the edges of any outerplanar graph G using χ(G) colors. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 22–36, 2010     

The determining number of a Cartesian product

A set S of vertices is a determining set for a graph G if every automorphism of G is uniquely determined by its action on S. The determining number of G, denoted Det(G), is the size of a smallest determining set. This paper begins by proving that if G=G□?□G is the prime factor decomposition of a connected graph then Det(G)=max{Det(G)}. It then provides upper and lower bounds for the determining number of a Cartesian power of a prime connected graph. Further, this paper shows that Det(Q_n)=?log₂n?+1 which matches the lower bound, and that Det(K)=?log₃(2n+1)?+1 which for all n is within one of the upper bound. The paper concludes by proving that if H is prime and connected, Det(Hⁿ)=Θ(logn). © 2009 Wiley Periodicals, Inc. J Graph Theory     

Balanced coloring of bipartite graphs

Given a bipartite graph G(U∪V, E) with n vertices on each side, an independent set I∈G such that |U∩I|=|V∩I| is called a balanced bipartite independent set. A balanced coloring of G is a coloring of the vertices of G such that each color class induces a balanced bipartite independent set in G. If graph G has a balanced coloring we call it colorable. The coloring number χ_B(G) is the minimum number of colors in a balanced coloring of a colorable graph G. We shall give bounds on χ_B(G) in terms of the average degree $\bar{d}$ of G and in terms of the maximum degree Δ of G. In particular we prove the following:

• $\chi_{{{B}}}({{G}}) \leq {{max}} \{{{2}},\lfloor {{2}}\overline{{{d}}}\rfloor+{{1}}\}$.
• For any 0<ε<1 there is a constant Δ₀ such that the following holds. Let G be a balanced bipartite graph with maximum degree Δ≥Δ₀ and n≥(1+ε)2Δ vertices on each side, then $\chi_{{{B}}}({{G}})\leq \frac{{{{20}}}}{\epsilon^{{{2}}}} \frac{\Delta}{{{{ln}}}\,\Delta}$.

© 2009 Wiley Periodicals, Inc. J Graph Theory 64: 277–291, 2010     

The size of strength-maximal graphs

Let G be a graph and let k′(G) be the edge-connectivity of G. The strength of G, denoted by k?′(G), is the maximum value of k′(H), where H runs over all subgraphs of G. A simple graph G is called k-maximal if k?′(G) ≤ k but for any edge e ∈ E(G^c), k?′(G + e) ≥ k + 1. Let G be a k-maximal graph of order n. In [3], Mader proved |E(G)| ≤ (n - k)k + (). In this note, we shall show (n - 1)k - () In?n/k + 2)? ≤ |E(G|, and characterize the extremal graphs. We shall also give a characterization of all k-maximal graphs.     

Total relative displacement of vertex permutations of K

Let α denote a permutation of the n vertices of a connected graph G. Define δ_α(G) to be the number , where the sum is over all the unordered pairs of distinct vertices of G. The number δ_α(G) is called the total relative displacement of α (in G). So, permutation α is an automorphism of G if and only if δ_α(G) = 0. Let π(G) denote the smallest positive value of δ_α(G) among the n! permutations α of the vertices of G. A permutation α for which π(G) = δ_α(G) has been called a near‐automorphism of G [ 2 ]. We determine π(K) and describe permutations α of K for which π(K) = δ_α(K). This is done by transforming the problem into the combinatorial optimization problem of maximizing the sums of the squares of the entries in certain t by t matrices with non–negative integer entries in which the sum of the entries in the ith row and the sum of the entries in the ith column each equal to n_i,1≤i≤t. We prove that for positive integers, n₁≤n₂≤…≤n_t, where t≥2 and n_t≥2, where k₀ is the smallest index for which n = n+1. As a special case, we correct the value of π(K_m,n), for all m and n at least 2, given by Chartrand, Gavlas, and VanderJagt [ 2 ]. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 85–100, 2002     

A neighborhood condition which implies the existence of a class of d-chromatic subgraphs

A graph G of order n satisfies the neighborhood condition NC_k > s if any collection of k independent vertices is collectively adjacent to more than s vertices. Given a family H of graphs, the decomposition class β(H) is the family of graphs B with the property that for some H ∈ H of chromatic number d, H contains B as an induced subgraph and l?V(H) ? V(B)? is (d ? 2) colorable. Let H be a family of d-chromatic graphs, B an element of β(H) such that B contains an s-matching as an induced subgraph. Thus the cardinality of one of the partite sets of B is s + r for some integer r ≥ 0. We show that if t is a fixed positive integer, G is a graph of sufficiently large order n that satisfies the neighborhood condition then G contains B + K(d - 2; t) as a subgraph.