Results on the Grundy chromatic number of graphs

Given a graph G, by a Grundy k-coloring of G we mean any proper k-vertex coloring of G such that for each two colors i and j, i<j, every vertex of G colored by j has a neighbor with color i. The maximum k for which there exists a Grundy k-coloring is denoted by Γ(G) and called Grundy (chromatic) number of G. We first discuss the fixed-parameter complexity of determining Γ(G)?k, for any fixed integer k and show that it is a polynomial time problem. But in general, Grundy number is an NP-complete problem. We show that it is NP-complete even for the complement of bipartite graphs and describe the Grundy number of these graphs in terms of the minimum edge dominating number of their complements. Next we obtain some additive Nordhaus-Gaddum-type inequalities concerning Γ(G) and Γ(G^c), for a few family of graphs. We introduce well-colored graphs, which are graphs G for which applying every greedy coloring results in a coloring of G with χ(G) colors. Equivalently G is well colored if Γ(G)=χ(G). We prove that the recognition problem of well-colored graphs is a coNP-complete problem.     

On the locating chromatic number of Kneser graphs

Let c be a proper k-coloring of a connected graph G and Π=(C₁,C₂,…,C_k) be an ordered partition of V(G) into the resulting color classes. For a vertex v of G, the color code of v with respect to Π is defined to be the ordered k-tuple c_Π(v):=(d(v,C₁),d(v,C₂),…,d(v,C_k)), where d(v,C_i)=min{d(v,x)|x∈C_i},1≤i≤k. If distinct vertices have distinct color codes, then c is called a locating coloring. The minimum number of colors needed in a locating coloring of G is the locating chromatic number of G, denoted by χ_L(G). In this paper, we study the locating chromatic number of Kneser graphs. First, among some other results, we show that χ_L(KG(n,2))=n−1 for all n≥5. Then, we prove that χ_L(KG(n,k))≤n−1, when n≥k². Moreover, we present some bounds for the locating chromatic number of odd graphs.     

Conditional colorings of graphs

For an integer r>0, a conditional(k,r)-coloring of a graph G is a proper k-coloring of the vertices of G such that every vertex of degree at least r in G will be adjacent to vertices with at least r different colors. The smallest integer k for which a graph G has a conditional (k,r)-coloring is the rth order conditional chromatic number χ_r(G). In this paper, the behavior and bounds of conditional chromatic number of a graph G are investigated.     

A note on list improper coloring of plane graphs

A list-assignment L to the vertices of G is an assignment of a set L(v) of colors to vertex v for every v∈V(G). An (L,d)^∗-coloring is a mapping ? that assigns a color ?(v)∈L(v) to each vertex v∈V(G) such that at most d neighbors of v receive color ?(v). A graph is called (k,d)^∗-choosable, if G admits an (L,d)^∗-coloring for every list assignment L with |L(v)|≥k for all v∈V(G). In this note, it is proved that every plane graph, which contains no 4-cycles and l-cycles for some l∈{8,9}, is (3,1)^∗-choosable.     

Greedy F-colorings of graphs

Let G=(V,E) be a connected graph. For a symmetric, integer-valued function δ on V×V, where K is an integer constant, N₀ is the set of nonnegative integers, and Z is the set of integers, we define a C-mapping by F(u,v,m)=δ(u,v)+m−K. A coloring c of G is an F-coloring if F(u,v,|c(u)−c(v)|)?0 for every two distinct vertices u and v of G. The maximum color assigned by c to a vertex of G is the value of c, and the F-chromatic number F(G) is the minimum value among all F-colorings of G. For an ordering of the vertices of G, a greedy F-coloring c of s is defined by (1) c(v₁)=1 and (2) for each i with 1?i<n, c(v_i+1) is the smallest positive integer p such that F(v_j,v_i+1,|c(v_j)−p|)?0, for each j with 1?j?i. The greedy F-chromatic number gF(s) of s is the maximum color assigned by c to a vertex of G. The greedy F-chromatic number of G is gF(G)=min{gF(s)} over all orderings s of V. The Grundy F-chromatic number is GF(G)=max{gF(s)} over all orderings s of V. It is shown that gF(G)=F(G) for every graph G and every F-coloring defined on G. The parameters gF(G) and GF(G) are studied and compared for a special case of the C-mapping F on a connected graph G, where δ(u,v) is the distance between u and v and .     

(3, 1)*-Choosability of graphs of nonnegative characteristic without intersecting short cycles

A graph G is called (k,d)^?-choosable if for every list assignment L satisfying ∣L(v)∣ ≥k for all v∈V(G), there is an L-coloring of G such that each vertex of G has at most d neighbors colored with the same color as itself. In this paper, it is proved that every graph of nonnegative characteristic without intersecting i-cycles for all i=3,4,5 is (3,1)^?-choosable.     

[r,s,t]-Coloring of K n,n

Let G=(V(G),E(G)) be a simple graph. Given non-negative integers r,s, and t, an [r,s,t]-coloring of G is a mapping c from V(G)∪E(G) to the color set {0,1,…,k?1} such that |c(v _i )?c(v _j )|≥r for every two adjacent vertices v _i ,v _j , |c(e _i )?c(e _j )|≥s for every two adjacent edges e _i ,e _j , and |c(v _i )?c(e _j )|≥t for all pairs of incident vertices and edges, respectively. The [r,s,t]-chromatic number χ _r,s,t (G) of G is defined to be the minimum k such that G admits an [r,s,t]-coloring. We determine χ _r,s,t (K _n,n ) in all cases.     

A note on list improper coloring planar graphs

A graph G is called (k, d)^*-choosable if, for every list assignment L satisfying |L(v)| = k for all v ϵ V(G), there is an L-coloring of G such that each vertex of G has at most d neighbors colored with the same color as itself. In this note, we prove that every planar graph without 4-cycles and l-cycles for some l ϵ {5, 6, 7} is (3, 1)^*-choosable.     

Acyclic colorings of graphs with bounded degree

A k-coloring (not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colors i and j the subgraph induced by the edges whose endpoints have colors i and j is acyclic. We consider some generalized acyclic k-colorings, namely, we require that each color class induces an acyclic or bounded degree graph. Mainly we focus on graphs with maximum degree 5. We prove that any such graph has an acyclic 5-coloring such that each color class induces an acyclic graph with maximum degree at most 4. We prove that the problem of deciding whether a graph G has an acyclic 2-coloring in which each color class induces a graph with maximum degree at most 3 is NP-complete, even for graphs with maximum degree 5. We also give a linear-time algorithm for an acyclic t-improper coloring of any graph with maximum degree d assuming that the number of colors is large enough.     

Some graft transformations and its applications on the distance spectral radius of a graph

Let D(G)=(d_i,j)_n×n denote the distance matrix of a connected graph G with order n, where d_ij is equal to the distance between v_i and v_j in G. The largest eigenvalue of D(G) is called the distance spectral radius of graph G, denoted by ?(G). In this paper, some graft transformations that decrease or increase ?(G) are given. With them, for the graphs with both order n and k pendant vertices, the extremal graphs with the minimum distance spectral radius are completely characterized; the extremal graph with the maximum distance spectral radius is shown to be a dumbbell graph (obtained by attaching some pendant edges to each pendant vertex of a path respectively) when 2≤k≤n−2; for k=1,2,3,n−1, the extremal graphs with the maximum distance spectral radius are completely characterized.     

Accretion centers: A generalization of branch weight centroids

An ordered n-tuple (v_i1,v_i2,…,v_in) is called a sequential labelling of graph G if {v_i1,v_i2,…,v_in} = V(G) and the subgraph induced by {v_i1,v_i2,…, v_ij} is connected for 1≤j≤n. Let σ(v;G) denote the number of sequential labellings of G with v_i1=v. Vertex v is defined to be an accretion center of G if σ is maximized at v. This is shown to generalize the concept of a branch weight centroid of a tree since a vertex in a tree is an accretion center if and only if it is a centroid vertex. It is not, however, a generalization of the concept of a median since for a general graph an accretion center is not necessarily a vertex of minimum distance. A method for computing σ(v;G) based upon edge contractions is described.     

deBruijn-like sequences and the irregular chromatic number of paths and cycles

A deBruijn sequence of orderk, or a k-deBruijn sequence, over an alphabet A is a sequence of length |A|^k in which the last element is considered adjacent to the first and every possible k-tuple from A appears exactly once as a string of k-consecutive elements in the sequence. We will say that a cyclic sequence is deBruijn-like if for some k, each of the consecutive k-element substrings is distinct.A vertex coloring χ:V(G)→[k] of a graph G is said to be proper if no pair of adjacent vertices in G receive the same color. Let C(v;χ) denote the multiset of colors assigned by a coloring χ to the neighbors of vertex v. A proper coloring χ of G is irregular if χ(u)=χ(v) implies that C(u;χ)≠C(v;χ). The minimum number of colors needed to irregularly color G is called the irregular chromatic number of G. The notion of the irregular chromatic number pairs nicely with other parameters aimed at distinguishing the vertices of a graph. In this paper, we demonstrate a connection between the irregular chromatic number of cycles and the existence of certain deBruijn-like sequences. We then determine the exact irregular chromatic number of C_n and P_n for n≥3, thus verifying two conjectures given by Okamoto, Radcliffe and Zhang.     

On Harary index of graphs

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. For a connected graph G=(V,E) and two nonadjacent vertices v_i and v_j in V(G) of G, recall that G+v_iv_j is the supergraph formed from G by adding an edge between vertices v_i and v_j. Denote the Harary index of G and G+v_iv_j by H(G) and H(G+v_iv_j), respectively. We obtain lower and upper bounds on H(G+v_iv_j)−H(G), and characterize the equality cases in those bounds. Finally, in this paper, we present some lower and upper bounds on the Harary index of graphs with different parameters, such as clique number and chromatic number, and characterize the extremal graphs at which the lower or upper bounds on the Harary index are attained.     

On periodicity of perfect colorings of the infinite hexagonal and triangular grids

A coloring of vertices of a graph G is called r-perfect, if the color structure of each ball of radius r in G depends only on the color of the center of the ball. The parameters of a perfect coloring are given by the matrix A = (a _ij ) _i,j=1 ⁿ , where n is the number of colors and a _ij is the number of vertices of color j in a ball centered at a vertex of color i. We study the periodicity of perfect colorings of the graphs of the infinite hexagonal and triangular grids. We prove that for every 1-perfect coloring of the infinite triangular and every 1- and 2-perfect coloring of the infinite hexagonal grid there exists a periodic perfect coloring with the same matrix. The periodicity of perfect colorings of big radii have been studied earlier.     

Simultaneous graph parameters: Factor domination and factor total domination

Let F₁,F₂,…,F_k be graphs with the same vertex set V. A subset S⊆V is a factor dominating set if in every F_i every vertex not in S is adjacent to a vertex in S, and a factor total dominating set if in every F_i every vertex in V is adjacent to a vertex in S. The cardinality of a smallest such set is the factor (total) domination number. In this note, we investigate bounds on the factor (total) domination number. These bounds exploit results on colorings of graphs and transversals of hypergraphs.     

A note on Roman domination in graphs

Let G=(V,E) be a simple graph. A subset S⊆V is a dominating set of G, if for any vertex u∈V-S, there exists a vertex v∈S such that uv∈E. The domination number of G, γ(G), equals the minimum cardinality of a dominating set. A Roman dominating function on graph G=(V,E) is a function f:V→{0,1,2} satisfying the condition that every vertex v for which f(v)=0 is adjacent to at least one vertex u for which f(u)=2. The weight of a Roman dominating function is the value f(V)=∑_v∈Vf(v). The Roman domination number of a graph G, denoted by γ_R(G), equals the minimum weight of a Roman dominating function on G. In this paper, for any integer k(2?k?γ(G)), we give a characterization of graphs for which γ_R(G)=γ(G)+k, which settles an open problem in [E.J. Cockayne, P.M. Dreyer Jr, S.M. Hedetniemi et al. On Roman domination in graphs, Discrete Math. 278 (2004) 11-22].     

Acyclic 6-choosability of planar graphs without adjacent short cycles

A proper vertex coloring of a graph G is acyclic if G contains no bicolored cycles.Given a list assignment L={L(v)|v∈V}of G,we say that G is acyclically L-colorable if there exists a proper acyclic coloringπof G such thatπ(v)∈L(v)for all v∈V.If G is acyclically L-colorable for any list assignment L with|L(v)|k for all v∈V(G),then G is acyclically k-choosable.In this paper,we prove that every planar graph G is acyclically 6-choosable if G does not contain 4-cycles adjacent to i-cycles for each i∈{3,4,5,6}.This improves the result by Wang and Chen(2009).     

Local condition for planar graphs of maximum degree 7 to be 8-totally colorable

The total chromatic number of a graph G, denoted by χ^″(G), is the minimum number of colors needed to color the vertices and edges of G such that no two adjacent or incident elements get the same color. It is known that if a planar graph G has maximum degree Δ≥9, then χ^″(G)=Δ+1. In this paper, we prove that if G is a planar graph with maximum degree 7, and for every vertex v, there is an integer k_v∈{3,4,5,6} so that v is not incident with any k_v-cycle, then χ^″(G)=8.     

Critically indecomposable graphs

An undirected graph G=(V,E) with a specific subset X⊂V is called X-critical if G and G(X), induced subgraph on X, are indecomposable but G(V−{w}) is decomposable for every w∈V−X. This is a generalization of critically indecomposable graphs studied by Schmerl and Trotter [J.H. Schmerl, W.T. Trotter, Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures, Discrete Mathematics 113 (1993) 191-205] and Bonizzoni [P. Bonizzoni, Primitive 2-structures with the (n−2)-property, Theoretical Computer Science 132 (1994) 151-178], who deal with the case where X is empty.We present several structural results for this class of graphs and show that in every X-critical graph the vertices of V−X can be partitioned into pairs (a₁,b₁),(a₂,b₂),…,(a_m,b_m) such that G(V−{a_j1,b_j1,…,a_jk,b_jk}) is also an X-critical graph for arbitrary set of indices {j₁,…,j_k}. These vertex pairs are called commutative elimination sequence. If G is an arbitrary indecomposable graph with an indecomposable induced subgraph G(X), then the above result establishes the existence of an indecomposability preserving sequence of vertex pairs (x₁,y₁),…,(x_t,y_t) such that x_i,y_i∈V−X. As an application of the commutative elimination sequence of an X-critical graph we present algorithms to extend a 3-coloring (similarly, 1-factor) of G(X) to entire G.