Vertex‐distinguishing edge colorings of graphs

We consider lower bounds on the the vertex‐distinguishing edge chromatic number of graphs and prove that these are compatible with a conjecture of Burris and Schelp 8 . We also find upper bounds on this number for certain regular graphs G of low degree and hence verify the conjecture for a reasonably large class of such graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 95–109, 2003     

图的D(2)-点可区别一般边染色

引入了图的D(β)-点可区别一般边染色,并对β=2的情形做了讨论,得到了路,圈,星,双星,扇,轮的D(2)-点可区别一般边色数,对于2距离色数等于3及4的图的D(2)-点可区别一般边色数做了探讨,特别研究了具有稳定2距离4着色的图的D(2)-点可区别一般边染色.文中提出了一个相关猜想和一个公开问题.     

Orientable edge colorings of graphs

An edge coloring of a graph is orientable if and only if it is possible to orient the edges of the graph so that the color of each edge is determined by the head of its corresponding oriented arc. The goals of this paper include finding a forbidden substructure characterization of orientable colorings and giving a linear time recognition algorithm for orientable colorings.An edge coloring is lexical if and only if it is possible to number the vertices of the graph so that the color of each edge is determined by its lower endpoint. Lexical colorings are, of course, the orientable colorings in which the underlying orientation is acyclic. Lexical colorings play an important role in Canonical Ramsey theory, and it is this standpoint that motivates the current study.     

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     

Gap vertex-distinguishing edge colorings of graphs

In this paper, we study a new coloring parameter of graphs called the gap vertex-distinguishing edge coloring. It consists in an edge-coloring of a graph $G$ which induces a vertex distinguishing labeling of $G$ such that the label of each vertex is given by the difference between the highest and the lowest colors of its adjacent edges. The minimum number of colors required for a gap vertex-distinguishing edge coloring of $G$ is called the gap chromatic number of $G$ and is denoted by $\mathrm{gap}\left(G\right)$.We here study the gap chromatic number for a large set of graphs $G$ of order $n$ and prove that $\mathrm{gap}\left(G\right)\in \{n?1,n,n+1\}$.     

Strong edge colorings of uniform graphs

For a graph G=(V(G),E(G)), a strong edge coloring of G is an edge coloring in which every color class is an induced matching. The strong chromatic index of G, χ_s(G), is the smallest number of colors in a strong edge coloring of G. The strong chromatic index of the random graph G(n,p) was considered in Discrete Math. 281 (2004) 129, Austral. J. Combin. 10 (1994) 97, Austral. J. Combin. 18 (1998) 219 and Combin. Probab. Comput. 11 (1) (2002) 103. In this paper, we consider χ_s(G) for a related class of graphs G known as uniform or ε-regular graphs. In particular, we prove that for 0<ε?d<1, all (d,ε)-regular bipartite graphs G=(U∪V,E) with |U|=|V|?n₀(d,ε) satisfy χ_s(G)?ζ(ε)Δ(G)², where ζ(ε)→0 as ε→0 (this order of magnitude is easily seen to be best possible). Our main tool in proving this statement is a powerful packing result of Pippenger and Spencer (Combin. Theory Ser. A 51(1) (1989) 24).     

Adjacent strong edge colorings and total colorings of regular graphs

It is conjectured that χas(G) = χt(G) for every k-regular graph G with no C5 component (k 2). This conjecture is shown to be true for many classes of graphs, including: graphs of type 1; 2-regular, 3-regular and (|V (G)| - 2)-regular graphs; bipartite graphs; balanced complete multipartite graphs; k-cubes; and joins of two matchings or cycles.     

Acyclic edge colorings of planar graphs and seriesparallel graphs

A proper edge coloring 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. Alon et al. conjectured that a′(G) ⩽ Δ(G) + 2 for any graphs. For planar graphs G with girth g(G), we prove that a′(G) ⩽ max{2Δ(G) − 2, Δ(G) + 22} if g(G) ⩾ 3, a′(G) ⩽ Δ(G) + 2 if g(G) ⩾ 5, a′(G) ⩽ Δ(G) + 1 if g(G) ⩾ 7, and a′(G) = Δ(G) if g(G) ⩾ 16 and Δ(G) ⩾ 3. For series-parallel graphs G, we have a′(G) ⩽ Δ(G) + 1. This work was supported by National Natural Science Foundation of China (Grant No. 10871119) and Natural Science Foundation of Shandong Province (Grant No. Y2008A20).     

r-Strong edge colorings of graphs

Let G be a graph and for any natural number r, denotes the minimum number of colors required for a proper edge coloring of G in which no two vertices with distance at most r are incident to edges colored with the same set of colors. In [Z. Zhang, L. Liu, J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett. 15 (2002) 623-626] it has been proved that for any tree T with at least three vertices, . Here we generalize this result and show that . Moreover, we show that if for any two vertices u and v with maximum degree d(u,v)?3, then . Also for any tree T with Δ(T)?3 we prove that . Finally, it is shown that for any graph G with no isolated edges, .     

On edge colorings of 1-toroidal graphs

A graph is 1-toroidal, if it can be embedded in the torus so that each edge is crossed by at most one other edge. In this paper, it is proved that every 1-toroidal graph with maximum degree Δ≥ 10 is of class one in terms of edge coloring. Meanwhile, we show that there exist class two 1-toroidal graphs with maximum degree Δ for each Δ≤ 8.     

Equitable edge‐colorings of simple graphs

An edge‐coloring of a graph G is equitable if, for each v∈V(G), the number of edges colored with any one color incident with v differs from the number of edges colored with any other color incident with v by at most one. A new sufficient condition for equitable edge‐colorings of simple graphs is obtained. This result covers the previous results, which are due to Hilton and de Werra, verifies a conjecture made by Hilton recently, and substantially extends it to a more general class of graphs. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:175‐197, 2011     

Neighbor sum distinguishing total colorings of triangle free planar graphs

A total k-coloring c of a graph G is a proper total coloring c of G using colors of the set[k] = {1, 2,..., k}. Let f(u) denote the sum of the color on a vertex u and colors on all the edges incident to u. A k-neighbor sum distinguishing total coloring of G is a total k-coloring of G such that for each edge uv ∈ E(G), f(u) = f(v). By χ nsd(G), we denote the smallest value k in such a coloring of G. Pil′sniak and Wo′zniak conjectured that χ nsd(G) ≤Δ(G) + 3 for any simple graph with maximum degree Δ(G). In this paper, by using the famous Combinatorial Nullstellensatz, we prove that the conjecture holds for any triangle free planar graph with maximum degree at least 7.     

Circular colorings of edge‐weighted graphs

The notion of (circular) colorings of edge‐weighted graphs is introduced. This notion generalizes the notion of (circular) colorings of graphs, the channel assignment problem, and several other optimization problems. For instance, its restriction to colorings of weighted complete graphs corresponds to the traveling salesman problem (metric case). It also gives rise to a new definition of the chromatic number of directed graphs. Several basic results about the circular chromatic number of edge‐weighted graphs are derived. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 107–116, 2003     

Asymmetric 2‐colorings of graphs

We show that the edges of every 3‐connected planar graph except K₄ can be colored with two colors in such a way that the graph has no color‐preserving automorphisms. Also, we characterize all graphs that have the property that their edges can be 2‐colored so that no matter how the graph is embedded in any orientable surface, there is no homeomorphism of the surface that induces a nontrivial color‐preserving automorphism of the graph.     

Neighbor sum distinguishing total colorings of K 4-minor free graphs

A total [k]-coloring of a graph G is a mapping φ： V（G） U E（G） →{1, 2, ..., k} such that any two adjacent elements in V（G）UE（G） receive different colors. Let f（v） denote the sum of the colors of a vertex v and the colors of all incident edges of v. A total [k]-neighbor sum distinguishing-coloring of G is a total [k]-coloring of G such that for each edge uv E E（G）, f（u） ≠ f（v）. By tt [G, Xsd（ J, we denote the smallest value k in such a coloring of G. Pilniak and Woniak conjectured X＇sd（G） 〈 A（G） ＋ 3 for any simple graph with maximum degree A（G）. This conjecture has been proved for complete graphs, cycles, bipartite graphs, and subcubic graphs. In this paper, we prove that it also holds for Ka-minor free graphs. Furthermore, we show that if G is a Ka-minor flee graph with A（G） 〉 4, then ＂ Xnsd（G） 〈 A（G） ＋ 2. The bound A（G） ＋ 2 is sharp.     

Some results on cyclic interval edge colorings of graphs

A proper edge coloring of a graph G with colors is called a cyclic interval t‐coloring if for each vertex v of G the edges incident to v are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. We prove that a bipartite graph G of even maximum degree admits a cyclic interval ‐coloring if for every vertex v the degree satisfies either or . We also prove that every Eulerian bipartite graph G with maximum degree at most eight has a cyclic interval coloring. Some results are obtained for ‐biregular graphs, that is, bipartite graphs with the vertices in one part all having degree a and the vertices in the other part all having degree b; it has been conjectured that all these have cyclic interval colorings. We show that all (4, 7)‐biregular graphs as well as all ‐biregular () graphs have cyclic interval colorings. Finally, we prove that all complete multipartite graphs admit cyclic interval colorings; this proves a conjecture of Petrosyan and Mkhitaryan.     

A note on transformations of edge colorings of bipartite graphs

The author and A. Mirumian proved the following theorem: Let G be a bipartite graph with maximum degree Δ and let t,n be integers, tnΔ. Then it is possible to obtain, from one proper edge t-coloring of G, any proper edge n-coloring of G using only transformations of 2-colored and 3-colored subgraphs such that the intermediate colorings are also proper. In this note we show that if t>Δ then we can transform f to g using only transformations of 2-colored subgraphs. We also correct the algorithm suggested in [A.S. Asratian, Short solution of Kotzig's problem for bipartite graphs, J. Combin. Theory Ser. B 74 (1998) 160–168] for transformation of f to g in the case when t=n=Δ and G is regular.     

Almost regular edge colorings and regular decompositions of complete graphs

For k = 1 and k = 2, we prove that the obvious necessary numerical conditions for packing t pairwise edge‐disjoint k‐regular subgraphs of specified orders m₁,m₂,… ,m_t in the complete graph of order n are also sufficient. To do so, we present an edge‐coloring technique which also yields new proofs of various known results on graph factorizations. For example, a new construction for Hamilton cycle decompositions of complete graphs is given. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 499–506, 2008