On Vizing’s bound for the chromatic index of a multigraph

Two of the basic results on edge coloring are Vizing’s Theorem [V.G. Vizing, On an estimate of the chromatic class of a p-graph, Diskret. Analiz. 3 (1964) 25-30 (in Russian); V.G. Vizing, The chromatic class of a multigraph, Kibernetika (Kiev) 3 (1965) 29-39 (in Russian). English translation in Cybernetics 1 32-41], which states that the chromatic index χ^′(G) of a (multi)graph G with maximum degree Δ(G) and maximum multiplicity μ(G) satisfies Δ(G)≤χ^′(G)≤Δ(G)+μ(G), and Holyer’s Theorem [I. Holyer, The NP-completeness of edge-colouring, SIAM J. Comput. 10 (1981) 718-720], which states that the problem of determining the chromatic index of even a simple graph is NP-hard. Hence, a good characterization of those graphs for which Vizing’s upper bound is attained seems to be unlikely. On the other hand, Vizing noticed in the first two above-cited references that the upper bound cannot be attained if Δ(G)=2μ(G)+1≥5. In this paper we discuss the problem: For which values Δ,μ does there exist a graph G satisfying Δ(G)=Δ, μ(G)=μ, and χ^′(G)=Δ+μ.     

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.     

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.     

Acyclic edge colorings of graphs

A proper coloring of the edges 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 a′(G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a′(G) ≥ Δ(G) + 2 where Δ(G) is the maximum degree in G. It is known that a′(G) ≤ 16 Δ(G) for any graph G. We prove that there exists a constant c such that a′(G) ≤ Δ(G) + 2 for any graph G whose girth is at least cΔ(G) log Δ(G), and conjecture that this upper bound for a′(G) holds for all graphs G. We also show that a′(G) ≤ Δ + 2 for almost all Δ‐regular graphs. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 157–167, 2001     

Injective coloring of planar graphs

A coloring of a graph G is injective if its restriction to the neighborhood of any vertex is injective. The injective chromatic numberχ_i(G) of a graph G is the least k such that there is an injective k-coloring. In this paper we prove that if G is a planar graph with girth g and maximum degree Δ, then (1) χ_i(G)=Δ if either g≥20 and Δ≥3, or g≥7 and Δ≥71; (2) χ_i(G)≤Δ+1 if g≥11; (3) χ_i(G)≤Δ+2 if g≥8.     

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.     

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.     

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.     

Acyclic chromatic indices of planar graphs with large girth

An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a^′(G) of G is the smallest k such that G has an acyclic edge coloring using k colors.In this paper, we prove that every planar graph G with girth g(G) and maximum degree Δ has a^′(G)=Δ if there exists a pair (k,m)∈{(3,11),(4,8),(5,7),(8,6)} such that G satisfies Δ≥k and g(G)≥m.     

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.     

Backbone colorings of graphs with bounded degree

We study backbone colorings, a variation on classical vertex colorings: Given a graph G and a subgraph H of G (the backbone of G), a backbone coloring for G and H is a proper vertex k-coloring of G in which the colors assigned to adjacent vertices in H differ by at least 2. The minimal k∈N for which such a coloring exists is called the backbone chromatic number of G. We show that for a graph G of maximum degree Δ where the backbone graph is a d-degenerated subgraph of G, the backbone chromatic number is at most Δ+d+1 and moreover, in the case when the backbone graph being a matching we prove that the backbone chromatic number is at most Δ+1. We also present examples where these bounds are attained.Finally, the asymptotic behavior of the backbone chromatic number is studied regarding the degrees of G and H. We prove for any sparse graph G that if the maximum degree of a backbone graph is small compared to the maximum degree of G, then the backbone chromatic number is at most .     

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.     

Total colorings of planar graphs without adjacent triangles

Let G be a planar graph without adjacent 3-cycles, that is, two cycles of length 3 are not incident with a common edge. In this paper, it is proved that the total coloring conjecture is true for G; moreover, if Δ(G)≥9, then the total chromatic number χ^″(G) of G is Δ(G)+1. Some other related results are obtained, too.     

Acyclic vertex coloring of graphs of maximum degree 5

An acyclic vertex coloring of a graph is a proper vertex coloring such that there are no bichromatic cycles. The acyclic chromatic number of G, denoted a(G), is the minimum number of colors required for acyclic vertex coloring of graph G. For a family F of graphs, the acyclic chromatic number of F, denoted by a(F), is defined as the maximum a(G) over all the graphs G∈F. In this paper we show that a(F)=8 where F is the family of graphs of maximum degree 5 and give a linear time algorithm to achieve this bound.     

Some sufficient conditions for a planar graph of maximum degree six to be Class 1

Let G be a planar graph of maximum degree 6. In this paper we prove that if G does not contain either a 6-cycle, or a 4-cycle with a chord, or a 5- and 6-cycle with a chord, then χ^′(G)=6, where χ^′(G) denotes the chromatic index of G.     

Locally restricted colorings

We consider the following problem: given suitable integers χ and p, what is the smallest value ρ such that, for any graph G with chromatic number χ and any vertex coloring of G with at most χ+p colors, there is a vertex v such that at least χ different colors occur within distance ρ of v? Let ρ(χ,p) be this value; we show in particular that ρ(χ,p)?⌈p/2⌉+1 for all χ,p. We give the exact value of ρ when p=0 or χ?3, and (χ,p)=(4,1) or (4,2).     

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.     

Total chromatic number of one kind of join graphs

The total chromatic number χ_T(G) of a graph G is the minimum number of colors needed to color the elements (vertices and edges) of G such that no adjacent or incident pair of elements receive the same color. G is called Type 1 if χ_T(G)=Δ(G)+1. In this paper we prove that the join of a complete inequibipartite graph K_n1,n2 and a path P_m is of Type 1.     

New upper bounds on linear coloring of planar graphs

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest number of colors in a linear coloring of G. In this paper, it is proved that every planar graph G with girth g and maximum degree Δ has(1)lc(G) ≤Δ 21 if Δ≥ 9; (2)lc(G) ≤「Δ/2」 + 7 ifg ≥ 5; (3) lc(G) ≤「Δ/2」 + 2 ifg ≥ 7 and Δ≥ 7.     

An upper bound for the adjacent vertex distinguishing acyclic edge chromatic number of a graph

A proper k-edge coloring of a graph G is called adjacent vertex distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the color set of edges incident to u is not equal to the color set of edges incident to υ, where uυ ∈ E(G). The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by χ′ _aa (G), is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. In this paper we prove that if G(V, E) is a graph with no isolated edges, then χ′ _aa (G) ≤ 32Δ. Supported by the Natural Science Foundation of Gansu Province (3ZS051-A25-025)