Local coloring of Kneser graphs

A local coloring of a graph G is a function c:V(G)→N having the property that for each set S⊆V(G) with 2≤|S|≤3, there exist vertices u,v∈S such that |c(u)−c(v)|≥m_S, where m_S is the number of edges of the induced subgraph 〈S〉. The maximum color assigned by a local coloring c to a vertex of G is called the value of c and is denoted by χ_?(c). The local chromatic number of G is χ_?(G)=min{χ_?(c)}, where the minimum is taken over all local colorings c of G. The local coloring of graphs was introduced by Chartrand et al. [G. Chartrand, E. Salehi, P. Zhang, On local colorings of graphs, Congressus Numerantium 163 (2003) 207-221]. In this paper the local coloring of Kneser graphs is studied and the local chromatic number of the Kneser graph K(n,k) for some values of n and k is determined.     

Linear choosability of graphs

A proper vertex coloring of a non-oriented graph G is linear if the graph induced by the vertices of any two color classes is a forest of paths. A graph G is linearly L-list colorable if for a given list assignment L={L(v):v∈V(G)}, there exists a linear coloring c of G such that c(v)∈L(v) for all v∈V(G). If G is linearly L-list colorable for any list assignment with |L(v)|?k for all v∈V(G), then G is said to be linearly k-choosable. In this paper, we investigate the linear choosability for some families of graphs: graphs with small maximum degree, with given maximum average degree, outerplanar and planar graphs. Moreover, we prove that deciding whether a bipartite subcubic planar graph is linearly 3-colorable is an NP-complete problem.     

Neighbor sum distinguishing total colorings via the Combinatorial Nullstellensatz

Let G=(V,E)be a graph andφbe a total coloring of G by using the color set{1,2,...,k}.Let f(v)denote the sum of the color of the vertex v and the colors of all incident edges of v.We say thatφis neighbor sum distinguishing if for each edge uv∈E(G),f(u)=f(v).The smallest number k is called the neighbor sum distinguishing total chromatic number,denoted byχ′′nsd(G).Pil′sniak and Wo′zniak conjectured that for any graph G with at least two vertices,χ′′nsd(G)(G)+3.In this paper,by using the famous Combinatorial Nullstellensatz,we show thatχ′′nsd(G)2(G)+col(G)-1,where col(G)is the coloring number of G.Moreover,we prove this assertion in its list version.     

Dynamic list coloring of bipartite graphs

A dynamic coloring of a graph is a proper coloring of its vertices such that every vertex of degree more than one has at least two neighbors with distinct colors. The least number of colors in a dynamic coloring of G, denoted by χ₂(G), is called the dynamic chromatic number of G. The least integer k, such that if every vertex of G is assigned a list of k colors, then G has a proper (resp. dynamic) coloring in which every vertex receives a color from its own list, is called the choice number of G, denoted by ch(G) (resp. the dynamic choice number, denoted by ch₂(G)). It was recently conjectured (Akbari et al. (2009) [1]) that for any graph G, ch₂(G)=max(ch(G),χ₂(G)). In this short note we disprove this conjecture. We first give an example of a small planar bipartite graph G with ch(G)=χ₂(G)=3 and ch₂(G)=4. Then, for any integer k≥5, we construct a bipartite graph G_k such that ch(G_k)=χ₂(G_k)=3 and ch₂(G)≥k.     

The strong chromatic index of a class of graphs

The strong chromatic index of a graph G is the minimum integer k such that the edge set of G can be partitioned into k induced matchings. Faudree et al. [R.J. Faudree, R.H. Schelp, A. Gyárfás, Zs. Tuza, The strong chromatic index of graphs, Ars Combin. 29B (1990) 205-211] proposed an open problem: If G is bipartite and if for each edge xy∈E(G), d(x)+d(y)≤5, then sχ^′(G)≤6. Let H₀ be the graph obtained from a 5-cycle by adding a new vertex and joining it to two nonadjacent vertices of the 5-cycle. In this paper, we show that if G (not necessarily bipartite) is not isomorphic to H₀ and d(x)+d(y)≤5 for any edge xy of G then sχ^′(G)≤6. The proof of the result implies a linear time algorithm to produce a strong edge coloring using at most 6 colors for such graphs.     

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.     

Concise proofs for adjacent vertex-distinguishing total colorings

Let G=(V,E) be a graph and f:(V∪E)→[k] be a proper total k-coloring of G. We say that f is an adjacent vertex- distinguishing total coloring if for any two adjacent vertices, the set of colors appearing on the vertex and incident edges are different. We call the smallest k for which such a coloring of G exists the adjacent vertex-distinguishing total chromatic number, and denote it by χ_at(G). Here we provide short proofs for an upper bound on the adjacent vertex-distinguishing total chromatic number of graphs of maximum degree three, and the exact values of χ_at(G) when G is a complete graph or a cycle.     

On DP-coloring of graphs and multigraphs

While solving a question on the list coloring of planar graphs, Dvo?ák and Postle introduced the new notion of DP-coloring (they called it correspondence coloring). A DP-coloring of a graph G reduces the problem of finding a coloring of G from a given list L to the problem of finding a “large” independent set in the auxiliary graph H(G,L) with vertex set {(v, c): v ∈ V (G) and c ∈ L(v)}. It is similar to the old reduction by Plesnevi? and Vizing of the k-coloring problem to the problem of finding an independent set of size |V(G)| in the Cartesian product G□K _k, but DP-coloring seems more promising and useful than the Plesnevi?–Vizing reduction. Some properties of the DP-chromatic number χ _DP (G) resemble the properties of the list chromatic number χ _l (G) but some differ quite a lot. It is always the case that χ _DP (G) ≥ χ _l (G). The goal of this note is to introduce DP-colorings for multigraphs and to prove for them an analog of the result of Borodin and Erd?s–Rubin–Taylor characterizing the multigraphs that do not admit DP-colorings from some DP-degree-lists. This characterization yields an analog of Gallai’s Theorem on the minimum number of edges in n-vertex graphs critical with respect to DP-coloring.     

The deficiency of a regular graph

A proper edge coloring c:E(G)→Z of a finite simple graph G is an interval coloring if the colors used at each vertex form a consecutive interval of integers. Many graphs do not have interval colorings, and the deficiency of a graph is an invariant that measures how close a graph comes to having an interval coloring. In this paper we search for tight upper bounds on the deficiencies of k-regular graphs in terms of the number of vertices. We find exact values for 1?k?4 and bounds for larger k.     

List coloring of graphs having cycles of length divisible by a given number

Let G be a graph and χ_l(G) denote the list chromatic number of G. In this paper we prove that for every graph G for which the length of each cycle is divisible by l (l≥3), χ_l(G)≤3.     

List total colorings of series-parallel graphs

A total coloring of a graph G is a coloring of all elements of G, i.e., vertices and edges, in such a way that no two adjacent or incident elements receive the same color. Let L(x) be a set of colors assigned to each element x of G. Then a list total coloring of G is a total coloring such that each element x receives a color contained in L(x). The list total coloring problem asks whether G has a list total coloring. In this paper, we first show that the list total coloring problem is NP-complete even for series-parallel graphs. We then give a sufficient condition for a series-parallel graph to have a list total coloring, that is, we prove a theorem that any series-parallel graph G has a list total coloring if |L(v)|min{5,Δ+1} for each vertex v and |L(e)|max{5,d(v)+1,d(w)+1} for each edge e=vw, where Δ is the maximum degree of G and d(v) and d(w) are the degrees of the ends v and w of e, respectively. The theorem implies that any series-parallel graph G has a total coloring with Δ+1 colors if Δ4. We finally present a linear-time algorithm to find a list total coloring of a given series-parallel graph G if G satisfies the sufficient condition.     

The total chromatic number of Pseudo-Halin graphs with lower degree

The total chromatic number χ_T(G) of a graph G is the least number of colors needed to color the vertices and the edges of G such that no adjacent or incident elements receive the same color. The Total Coloring Conjecture(TCC) states that for every simple graph G, χ_T(G)≤Δ(G)+2. In this paper, we show that χ_T(G)=Δ(G)+1 for all pseudo-Halin graphs with Δ(G)=4 and 5.     

On the acyclic choosability of graphs

A proper vertex coloring of a graph G = (V,E) is acyclic if G contains no bicolored cycle. A graph G is L‐list colorable if for a given list assignment L = {L(v): v ∈ V}, there exists a proper coloring c of G such that c (v) ∈ L(v) for all v ∈ V. If G is L‐list colorable for every list assignment with |L (v)| ≥ k for all v ∈ V, then G is said k‐choosable. A graph is said to be acyclically k‐choosable if the obtained coloring is acyclic. In this paper, we study the links between acyclic k‐choosability of G and Mad(G) defined as the maximum average degree of the subgraphs of G and give some observations about the relationship between acyclic coloring, choosability, and acyclic choosability. © 2005 Wiley Periodicals, Inc. J Graph Theory 51: 281–300, 2006     

On the packing chromatic number of Cartesian products, hexagonal lattice, and trees

The packing chromatic number χ_ρ(G) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into packings with pairwise different widths. Several lower and upper bounds are obtained for the packing chromatic number of Cartesian products of graphs. It is proved that the packing chromatic number of the infinite hexagonal lattice lies between 6 and 8. Optimal lower and upper bounds are proved for subdivision graphs. Trees are also considered and monotone colorings are introduced.     

On the Acyclic Chromatic Number of Hamming Graphs

An acyclic coloring of a graph G is a proper coloring of the vertex set of G such that G contains no bichromatic cycles. The acyclic chromatic number of a graph G is the minimum number k such that G has an acyclic coloring with k colors. In this paper, acyclic colorings of Hamming graphs, products of complete graphs, are considered. Upper and lower bounds on the acyclic chromatic number of Hamming graphs are given. Gretchen L. Matthews: The work of this author is supported by NSA H-98230-06-1-0008.     

Every toroidal graph is acyclically 8-choosable

A proper coloring of a graphG is acyclic if G contains no 2-colored cycle.A graph G is acyclically L-list colorable if for a given list assignment L={L(v):v∈V(G)},there exists a proper acyclic coloringφof G such thatφ(v)∈L(v)for all v∈V(G).If G is acyclically L-list colorable for any list assignment L with|L(v)|≥k for all v∈V(G),then G is acyclically k-choosable.In this article,we prove that every toroidal graph is acyclically 8-choosable.     

Acyclic 5-choosability of planar graphs without 4-cycles

A proper vertex coloring of a graph G=(V,E) is acyclic if G contains no bicolored cycle. A graph G is acyclically L-list colorable if for a given list assignment L={L(v):v∈V}, there exists a proper acyclic coloring π of G such that π(v)∈L(v) for all v∈V. If G is acyclically L-list colorable for any list assignment with |L(v)|≥k for all v∈V, then G is acyclically k-choosable. In this paper we prove that every planar graph without 4-cycles and without two 3-cycles at distance less than 3 is acyclically 5-choosable. This improves a result in [M. Montassier, P. Ochem, A. Raspaud, On the acyclic choosability of graphs, J. Graph Theory 51 (2006) 281-300], which says that planar graphs of girth at least 5 are acyclically 5-choosable.     

On the adjacent vertex-distinguishing equitable edge coloring of graphs

Let G(V, E) be a graph. A k-adjacent vertex-distinguishing equatable edge coloring of G, k-AVEEC for short, is a proper edge coloring f if (1) C(u)≠C(v) for uv ∈ E(G), where C(u) = {f(uv)|uv ∈ E}, and (2) for any i, j = 1, 2,… k, we have ||Ei| |Ej|| ≤ 1, where Ei = {e|e ∈ E(G) and f(e) = i}. χáve (G) = min{k| there exists a k-AVEEC of G} is called the adjacent vertex-distinguishing equitable edge chromatic number of G. In this paper, we obtain the χáve (G) of some special graphs and present a conjecture.     

Chromatic coloring with a maximum color class

Let G be any graph, and also let Δ(G), χ(G) and α(G) denote the maximum degree, the chromatic number and the independence number of G, respectively. A chromatic coloring of G is a proper coloring of G using χ(G) colors. A color class in a proper coloring of G is maximum if it has size α(G). In this paper, we prove that if a graph G (not necessarily connected) satisfies χ(G)≥Δ(G), then there exists a chromatic coloring of G in which some color class is maximum. This cannot be guaranteed if χ(G)<Δ(G). We shall also give some other extensions.     

Colorings at minimum cost

We define by min_c∑_{u,v}∈E(G)|c(u)−c(v)| the min-costMC(G) of a graph G, where the minimum is taken over all proper colorings c. The min-cost-chromatic numberχ_M(G) is then defined to be the (smallest) number of colors k for which there exists a proper k-coloring c attaining MC(G). We give constructions of graphs G where χ(G) is arbitrarily smaller than χ_M(G). On the other hand, we prove that for every 3-regular graph G^′, χ_M(G^′)≤4 and for every 4-regular line graph G^″, χ_M(G^″)≤5. Moreover, we show that the decision problem whether χ_M(G)=k is -hard for k≥3.