Injective colorings of sparse graphs

Let denote the maximum average degree (over all subgraphs) of G and let χ_i(G) denote the injective chromatic number of G. We prove that if , then χ_i(G)≤Δ(G)+1; and if , then χ_i(G)=Δ(G). Suppose that G is a planar graph with girth g(G) and Δ(G)≥4. We prove that if g(G)≥9, then χ_i(G)≤Δ(G)+1; similarly, if g(G)≥13, then χ_i(G)=Δ(G).     

On signed cycle domination in graphs

Let G=(V,E) be a graph, a function f:E→{−1,1} is said to be an signed cycle dominating function (SCDF) of G if ∑_e∈E(C)f(e)≥1 holds for any induced cycle C of G. The signed cycle domination number of G is defined as is an SCDF of G}. In this paper, we obtain bounds on , characterize all connected graphs G with , and determine the exact value of for some special classes of graphs G. In addition, we pose some open problems and conjectures.     

Long cycles and paths in distance graphs

For n∈N and D⊆N, the distance graph has vertex set {0,1,…,n−1} and edge set {ij∣0≤i,j≤n−1,|j−i|∈D}. Note that the important and very well-studied circulant graphs coincide with the regular distance graphs.A fundamental result concerning circulant graphs is that for these graphs, a simple greatest common divisor condition, their connectivity, and the existence of a Hamiltonian cycle are all equivalent. Our main result suitably extends this equivalence to distance graphs. We prove that for a finite set D of order at least 2, there is a constant c_D such that the greatest common divisor of the integers in D is 1 if and only if for every n, has a component of order at least n−c_D if and only if for every n≥c_D+3, has a cycle of order at least n−c_D. Furthermore, we discuss some consequences and variants of this result.     

Bounds on isoperimetric values of trees

Let G=(V,E) be a finite, simple and undirected graph. For S⊆V, let δ(S,G)={(u,v)∈E:u∈S and v∈V−S} be the edge boundary of S. Given an integer i, 1≤i≤|V|, let the edge isoperimetric value of G at i be defined as b_e(i,G)=min_S⊆V;|S|=i|δ(S,G)|. The edge isoperimetric peak of G is defined as b_e(G)=max_1≤j≤|V|b_e(j,G). Let b_v(G) denote the vertex isoperimetric peak defined in a corresponding way. The problem of determining a lower bound for the vertex isoperimetric peak in complete t-ary trees was recently considered in [Y. Otachi, K. Yamazaki, A lower bound for the vertex boundary-width of complete k-ary trees, Discrete Mathematics, in press (doi:10.1016/j.disc.2007.05.014)]. In this paper we provide bounds which improve those in the above cited paper. Our results can be generalized to arbitrary (rooted) trees.The depth d of a tree is the number of nodes on the longest path starting from the root and ending at a leaf. In this paper we show that for a complete binary tree of depth d (denoted as ), and where c₁, c₂ are constants. For a complete t-ary tree of depth d (denoted as ) and d≥clogt where c is a constant, we show that and where c₁, c₂ are constants. At the heart of our proof we have the following theorem which works for an arbitrary rooted tree and not just for a complete t-ary tree. Let T=(V,E,r) be a finite, connected and rooted tree — the root being the vertex r. Define a weight function w:V→N where the weight w(u) of a vertex u is the number of its successors (including itself) and let the weight index η(T) be defined as the number of distinct weights in the tree, i.e η(T)=|{w(u):u∈V}|. For a positive integer k, let ?(k)=|{i∈N:1≤i≤|V|,b_e(i,G)≤k}|. We show that .     

Roman domination in regular graphs

A Roman domination function on a graph G=(V(G),E(G)) is a function f:V(G)→{0,1,2} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. The weight of a Roman dominating function is the value f(V(G))=∑_u∈V(G)f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. Cockayne et al. [E. J. Cockayne et al. Roman domination in graphs, Discrete Mathematics 278 (2004) 11-22] showed that γ(G)≤γ_R(G)≤2γ(G) and defined a graph G to be Roman if γ_R(G)=2γ(G). In this article, the authors gave several classes of Roman graphs: P_3k,P_3k+2,C_3k,C_3k+2 for k≥1, K_m,n for min{m,n}≠2, and any graph G with γ(G)=1; In this paper, we research on regular Roman graphs and prove that: (1) the circulant graphs and , n⁄≡1 (mod (2k+1)), (n≠2k) are Roman graphs, (2) the generalized Petersen graphs P(n,2k+1)( (mod 4) and ), P(n,1) (n⁄≡2 (mod 4)), P(n,3) ( (mod 4)) and P(11,3) are Roman graphs, and (3) the Cartesian product graphs are Roman graphs.     

Generalization of transitive fraternal augmentations for directed graphs and its applications

Let be a directed graph. A transitive fraternal augmentation of is a directed graph with the same vertex set, including all the arcs of and such that for any vertices x,y,z,

1.

if and then or (fraternity);

2.

if and then (transitivity).

In this paper, we explore some generalization of the transitive fraternal augmentations for directed graphs and its applications. In particular, we show that the 2-coloring number col₂(G)≤O(∇₁(G)∇₀(G)²), where ∇_k(G) (k≥0) denotes the greatest reduced average density with depth k of a graph G; we give a constructive proof that ∇_k(G) bounds the distance (k+1)-coloring number col_k+1(G) with a function f(∇_k(G)). On the other hand, ∇_k(G)≤(col_2k+1(G))^2k+1. We also show that an inductive generalization of transitive fraternal augmentations can be used to study nonrepetitive colorings of graphs.     

Graham’s pebbling conjecture on product of thorn graphs of complete graphs

The pebbling number of a graph G, f(G), is the least n such that, no matter how n pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of pebbling moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. Let p₁,p₂,…,p_n be positive integers and G be such a graph, V(G)=n. The thorn graph of the graph G, with parameters p₁,p₂,…,p_n, is obtained by attaching p_i new vertices of degree 1 to the vertex u_i of the graph G, i=1,2,…,n. Graham conjectured that for any connected graphs G and H, f(G×H)≤f(G)f(H). We show that Graham’s conjecture holds true for a thorn graph of the complete graph with every by a graph with the two-pebbling property. As a corollary, Graham’s conjecture holds when G and H are the thorn graphs of the complete graphs with every .     

On global domination critical graphs

A dominating set of a graph G=(V,E) is a subset S⊆V such that every vertex not in S is adjacent to at least one vertex of S. The domination number of G is the cardinality of a smallest dominating set. The global domination number, γ_g(G), is the cardinality of a smallest set S that is simultaneously a dominating set of both G and its complement . Graphs for which γ_g(G−e)>γ_g(G) for all edges e∈E are characterized, as are graphs for which γ_g(G−e)<γ_g(G) for all edges e∈E whenever is disconnected. Progress is reported in the latter case when is connected.     

Lattices generated by join of strongly closed subgraphs in d-bounded distance-regular graphs

Let Γ be a d-bounded distance-regular graph with diameter d?3. Suppose that P(x) is a set of all strongly closed subgraphs containing x and that P(x,i) is a subset of P(x) consisting of all elements of P(x) with diameter i. Let L^′(x,i) be the set generated by all joins of the elements in P(x,i). By ordering L^′(x,i) by inclusion or reverse inclusion, L^′(x,i) is denoted by or . We prove that and are both finite atomic lattices, and give the conditions for them both being geometric lattices. We also give the eigenpolynomial of      

Directed 3-cycle decompositions of complete directed graphs with quadratic leaves

Let G be a vertex-disjoint union of directed cycles in the complete directed graph D_t, let |E(G)| be the number of directed edges of G and suppose or if t=5, and if t=6. It is proved in this paper that for each positive integer t, there exist -decompositions for D_t−G if and only if .     

The asymptotic number of spanning trees in circulant graphs

Let T(G) be the number of spanning trees in graph G. In this note, we explore the asymptotics of T(G) when G is a circulant graph with given jumps.The circulant graph is the 2k-regular graph with n vertices labeled 0,1,2,…,n−1, where node i has the 2k neighbors i±s₁,i±s₂,…,i±s_k where all the operations are . We give a closed formula for the asymptotic limit as a function of s₁,s₂,…,s_k. We then extend this by permitting some of the jumps to be linear functions of n, i.e., letting s_i, d_i and e_i be arbitrary integers, and examining

     

The orientable genus of some joins of complete graphs with large edgeless graphs

In an earlier paper the authors showed that with one exception the nonorientable genus of the graph with m≥n−1, the join of a complete graph with a large edgeless graph, is the same as the nonorientable genus of the spanning subgraph . The orientable genus problem for with m≥n−1 seems to be more difficult, but in this paper we find the orientable genus of some of these graphs. In particular, we determine the genus of when n is even and m≥n, the genus of when n=2^p+2 for p≥3 and m≥n−1, and the genus of when n=2^p+1 for p≥3 and m≥n+1. In all of these cases the genus is the same as the genus of K_m,n, namely ⌈(m−2)(n−2)/4⌉.     

Perfectness and imperfectness of unit disk graphs on triangular lattice points

Given a finite set of 2-dimensional points P⊆R² and a positive real d, a unit disk graph, denoted by (P,d), is an undirected graph with vertex set P such that two vertices are adjacent if and only if the Euclidean distance between the pair is less than or equal to d. Given a pair of non-negative integers m and n, P(m,n) denotes a subset of 2-dimensional triangular lattice points defined by where . Let T_m,n(d) be a unit disk graph defined on a vertex set P(m,n) and a positive real d. Let be the kth power of T_m,n(1).In this paper, we show necessary and sufficient conditions that [ is perfect] and/or [ is perfect], respectively. These conditions imply polynomial time approximation algorithms for multicoloring (T_m,n(d),w) and .     

On Seymour’s strengthening of Hadwiger’s conjecture for graphs with certain forbidden subgraphs

Let H be a set of graphs. A graph is called H-free if it does not contain a copy of a member of H as an induced subgraph. If H is a graph then G is called H-free if it is {H}-free. Plummer, Stiebitz, and Toft proved that, for every -free graph H on at most four vertices, every -free graph G has a collection of ⌈|V(G)|/2⌉ many pairwise adjacent vertices and edges (where a vertexvand an edgeeare adjacent if v is disjoint from the set V(e) of endvertices of e and adjacent to some vertex of V(e), and two edgeseandfare adjacent if V(e) and V(f) are disjoint and some vertex of V(e) is adjacent to some vertex of V(f)). Here we generalize this statement to -free graphs H on at most five vertices.     

Further results on the enumeration of hamilton paths in Cayley digraphs on semidirect products of cyclic groups

First, let m and n be positive integers such that n is odd and gcd(m,n)=1. Let G be the semidirect product of cyclic groups given by . Then the number of hamilton paths in Cay(G:x,y) (with initial vertex 1) is one fewer than the number of visible lattice points that lie on the closed quadrilateral whose vertices in consecutive order are (0,0), (4mn²+2n,16m²n), (n,4m), and (0,8m). Second, let m and n be positive integers such that n is odd. Let G be the semidirect product of cyclic groups given by . Then the number of hamilton paths in Cay(G:x,y) (with initial vertex 1) is (3m-1)n+m⌊(n+1)/3⌋+1.     

Boxicity of line graphs

The boxicity of a graph H, denoted by , is the minimum integer k such that H is an intersection graph of axis-parallel k-dimensional boxes in R^k. In this paper we show that for a line graph G of a multigraph, , where Δ(G) denotes the maximum degree of G. Since G is a line graph, Δ(G)≤2(χ(G)−1), where χ(G) denotes the chromatic number of G, and therefore, . For the d-dimensional hypercube Q_d, we prove that . The question of finding a nontrivial lower bound for was left open by Chandran and Sivadasan in [L. Sunil Chandran, Naveen Sivadasan, The cubicity of Hypercube Graphs. Discrete Mathematics 308 (23) (2008) 5795–5800].The above results are consequences of bounds that we obtain for the boxicity of a fully subdivided graph (a graph that can be obtained by subdividing every edge of a graph exactly once).     

Acyclic edge colouring of planar graphs without short cycles

Let G=(V,E) be any finite graph. A mapping C:E→[k] is called an acyclic edgek-colouring of G, if any two adjacent edges have different colours and there are no bichromatic cycles in G. In other words, for every pair of distinct colours i and j, the subgraph induced in G by all the edges which have colour i or j, is acyclic. The smallest number k of colours, such that G has an acyclic edge k-colouring is called the acyclic chromatic index of G, denoted by .In 2001, Alon et al. conjectured that for any graph G it holds that ; here Δ(G) stands for the maximum degree of G.In this paper we prove this conjecture for planar graphs with girth at least 5 and for planar graphs not containing cycles of length 4,6,8 and 9. We also show that if G is planar with girth at least 6. Moreover, we find an upper bound for the acyclic chromatic index of planar graphs without cycles of length 4. Namely, we prove that if G is such a graph, then .