We prove that the circular chromatic index of a cubic graph G with 2k vertices and chromatic index 4 is at least 3 + 2/k. This bound is (asymptotically) optimal for an infinite class of cubic graphs containing bridges. We also show that the constant 2 in the above bound can be increased for graphs with larger girth or higher connectivity. In particular, if G has girth at least 5, its circular c...

The integer round-up 4(G) of the fractional chromatic index yields the standard lower bound for the chromatic index of a multigraph G. We show that if G has even order n, then the chromatic index exceeds 4(G) by at most max{log,,, n, 1 + n/30}. More generally, we show that for any real b, 2/3 <b < 1, the chromatic index of G exceeds 4(G) by at most max{log,,b n, 1 +n(l b)/lO}. This is used to s...

Let G be a graph with n nodes, e edges, chromatic number and girth g. In an acyclic orientation of G, an arc is dependent if its reversal creates a cycle. It is well known that if < g, then G has an acyclic orientation without dependent arcs. Edelman showed that if G is connected, then every acyclic orientation has at most e ? n + 1 dependent arcs. We show that if G is connected and < g, then G...

We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, and simplicial complexes. This coloring also generalizes oriented coloring, acyclic coloring, and star coloring. There is an associated symmetric function in noncommuting variables for which we give a deletion-contraction formula. In the case of graphs this symmetric function in noncommuting variab...

For graphs of bounded maximum degree, we consider acyclic t-improper colourings, that is, colourings in which each bipartite subgraph consisting of the edges between two colour classes is acyclic and each colour class induces a graph with maximum degree at most t. In the first part, we show that all subcubic graphs are acyclically 1-improperly 3-choosable, thus extending a result of Boiron, Sop...

A vertex colouring of a graph G is called acyclic if no two adjacent vertices have the same colour and there is no two-coloured cycle in G. The acyclic chromatic number of G, denoted by A(G), is the least number of colours in an acyclic colouring of G. We show that if G has maximum degree d then A(G) = O(d 4 3 ) as d → ∞. This settles a problem of Erdős who conjectured, in 1976, that A(G) = o(d...

A coloring of a graph G is an assignment of colors to the vertices of G such that any two vertices receive distinct colors whenever they are adjacent. An acyclic coloring of G is a coloring such that no cycle of G receives exactly two colors, and the acyclic chromatic number χA(G) of a graph G is the minimum number of colors in any such coloring of G. Given a graph G and an integer k, determini...

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) ≥ ∆(G) + 2 where ∆(G) is the maximum degree in G. It is known that a′(G) ≤ 16∆(G) for any graph G (see [2],[10]). We prove that there exists a const...

A graph G is chordless if no cycle in G has a chord. In the present work we investigate the chromatic index and total chromatic number of chordless graphs. We describe a known decomposition result for chordless graphs and use it to establish that every chordless graph of maximum degree ∆ ≥ 3 has chromatic index ∆ and total chromatic number ∆+1. The proofs are algorithmic in the sense that we ac...

Properties of the 11 084 874 829 Steiner triple systems of order 19 are examined. In particular, there is exactly one 5-sparse, but no 6-sparse, STS(19); there is exactly one uniform STS(19); there are exactly two STS(19) with no almost parallel classes; all STS(19) have chromatic number 3; all have chromatic index 10, except for 4 075 designs with chromatic index 11 and two with chromatic inde...