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.     

The Maximum Chromatic Index of Multigraphs with Given Δ and?μ

Let f(Δ,?μ) =?max {χ′(G) | Δ (G) =?Δ,?μ(G) =?μ} where χ′(G), Δ(G) and?μ(G) denote the the chromatic index, the maximum degree and the maximum multiplicity of the multigraph G, respectively. If Δ < 2μ, then Shannon’s bound implies that the gap between f(Δ,?μ) and Vizing’s bound Δ +?μ can be arbitrarily large. In this note, we prove that this is also the case for Δ ≥?2μ (see Theorem 4).     

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.     

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).     

Achieving maximum chromatic index in multigraphs

Let G be a multigraph with maximum degree Δ and maximum edge multiplicity μ. Vizing’s Theorem says that the chromatic index of G is at most Δ+μ. If G is bipartite its chromatic index is well known to be exactly Δ. Otherwise G contains an odd cycle and, by a theorem of Goldberg, its chromatic index is at most , where g_o denotes odd-girth. Here we prove that a connected G achieves Goldberg’s upper bound if and only if G=μC_go and (g_o−1)∣2(μ−1). The question of whether or not G achieves Vizing’s upper bound is NP-hard for μ=1, but for μ≥2 we have reason to believe that this may be answerable in polynomial time. We prove that, with the exception of μK₃, every connected G with μ≥2 which achieves Vizing’s upper bound must contain a specific dense subgraph on five vertices. Additionally, if Δ≤μ², we prove that G must contain K₅, so G must be nonplanar. These results regarding Vizing’s upper bound extend work by Kierstead, whose proof technique influences us greatly here.     

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.     

Connectivity of the Mycielskian of a graph

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), which is called the Mycielskian of G. This paper investigates the vertex-connectivity κ(μ(G)) and edge-connectivity κ^′(μ(G)) of μ(G) . We show that κ(μ(G))=min{δ(μ(G)),2κ(G)+1} and κ^′(μ(G))=δ(μ(G)).     

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.     

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.     

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.     

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.     

On the chromatic index of multigraphs without large triangles

Let M be a multigraph. Vizing (Kibernetika (Kiev)1 (1965), 29–39) proved that χ′(M)≤Δ(M)+μ(M). Here it is proved that if χ′(M)≥Δ(M)+s, where $\text{1}\text{2}\text{(μ(M) + 1) < s}$ then M contains a 2s-sided triangle. In particular, (C′) if μ(M)≤2 and M does not contain a 4-sided triangle then χ′(M)≤Δ(M) + 1. Javedekar (J. Graph Theory4 (1980), 265–268) had conjectured that (C) if G is a simple graph that does not induce K_1,3 or K₅?e then χ(G)≤ω(G) + 1. The author and Schmerl (Discrete Math.45 (1983), 277–285) proved that (C′) implies (C); thus Javedekar's conjecture is true.     

Planar graphs without 5-cycles or without 6-cycles

Let G be a planar graph without 5-cycles or without 6-cycles. In this paper, we prove that if G is connected and δ(G)≥2, then there exists an edge xy∈E(G) such that d(x)+d(y)≤9, or there is a 2-alternating cycle. By using the above result, we obtain that (1) its linear 2-arboricity , (2) its list total chromatic number is Δ(G)+1 if Δ(G)≥8, and (3) its list edge chromatic number is Δ(G) if Δ(G)≥8.     

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.     

Improvement on Brooks' chromatic bound for a class of graphs

Brooks' Theorem says that if for a graph G,Δ(G)=n, then G is n-colourable, unless (1) n=2 and G has an odd cycle as a component, or (2) n>2 and K_n+1 is a component of G. In this paper we prove that if a graph G has none of some three graphs (K_1,3;K₅?e and H) as an induced subgraph and if Δ(G)?6 and d(G)<Δ(G), then χ(G)<Δ(G). Also we give examples to show that the hypothesis Δ(G)?6 can not be non-trivially relaxed and the graph K₅?e can not be removed from the hypothesis. Moreover, for a graph G with none of K_1,3;K₅?e and H as an induced subgraph, we verify Borodin and Kostochka's conjecture that if for a graph G,Δ(G)?9 and d(G)<Δ(G), then χ(G)<Δ(G).     

A note on acyclic edge coloring of complete bipartite graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic (2-colored) cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a^′(G). Let Δ=Δ(G) denote the maximum degree of a vertex in a graph G. A complete bipartite graph with n vertices on each side is denoted by K_n,n. Alon, McDiarmid and Reed observed that a^′(K_p−1,p−1)=p for every prime p. In this paper we prove that a^′(K_p,p)≤p+2=Δ+2 when p is prime. Basavaraju, Chandran and Kummini proved that a^′(K_n,n)≥n+2=Δ+2 when n is odd, which combined with our result implies that a^′(K_p,p)=p+2=Δ+2 when p is an odd prime. Moreover we show that if we remove any edge from K_p,p, the resulting graph is acyclically Δ+1=p+1-edge-colorable.     

Independence and coloring properties of direct products of some vertex-transitive graphs

Let α(G) and χ(G) denote the independence number and chromatic number of a graph G, respectively. Let G×H be the direct product graph of graphs G and H. We show that if G and H are circular graphs, Kneser graphs, or powers of cycles, then α(G×H)=max{α(G)|V(H)|,α(H)|V(G)|} and χ(G×H)=min{χ(G),χ(H)}.     

Vizing’s conjecture for chordal graphs

Vizing conjectured that γ(G□H)≥γ(G)γ(H) for every pair G,H of graphs, where “□” is the Cartesian product, and γ(G) is the domination number of the graph G. Denote by γⁱ(G) the maximum, over all independent sets I in G, of the minimal number of vertices needed to dominate I. We prove that γ(G□H)≥γⁱ(G)γ(H). Since for chordal graphs γⁱ=γ, this proves Vizing’s conjecture when G is chordal.     

Edge irregular total labellings for graphs of linear size

As an edge variant of the well-known irregularity strength of a graph G=(V,E) we investigate edge irregular total labellings, i.e. functions f:V∪E→{1,2,…,k} such that f(u)+f(uv)+f(v)≠f(u^′)+f(u^′v^′)+f(v^′) for every pair of different edges uv,u^′v^′∈E. The smallest possible k is the total edge irregularity strength of G. Confirming a conjecture by Ivan?o and Jendrol’ for a large class of graphs we prove that the natural lower bound is tight for every graph of order n, size m and maximum degree Δ with m>111000Δ. This also implies that the probability that a random graph from G(n,p(n)) satisfies the Ivan?o-Jendrol’ Conjecture tends to 1 as n→∞ for all functions p∈[0,1]^N. Furthermore, we prove that is an upper bound for every graph G of order n and size m≥3 whose edges are not all incident to a single vertex.