Let G be a directed graph, and let ∆ACY G be the simplicial complex whose simplices are the edge sets of acyclic subgraphs of G. Similarly, we define ∆NSC G to be the simplicial complex with the edge sets of not strongly connected subgraphs of G as simplices. We show that ∆ACY G is homotopy equivalent to the (n−1−k)-dimensional sphere if G is a disjoint union of k strongly connected graphs. Oth...

A strong edge coloring of a graph is a proper edge coloring in which every color class is an induced matching. The strong chromatic index of a graph is the minimum number of colors needed to obtain a strong edge coloring. In an analogous way, we can define the list version of strong edge coloring and list version of strong chromatic index. In this paper we prove that if G is a graph with maximu...

We shall use the distance chromatic index defined by the present author in early nineties, cf. [5] or [4] of 1993. The edge distance of two edges in a multigraph M is defined to be their distance in the line graph L(M) of M . Given a positive integer d, define the d+-chromatic index of the multigraph M , denoted by q(d)(M), to be equal to the chromatic number χ of the dth power of the line grap...

Our motivation is the question how similar the f -colouring problem is to the classic edge-colouring problem, particularly with regard to graph parameters. In 2010, Zhang, Yu, and Liu [9] gave a new description of the f -matching polytope and derived a formula for the fractional f -chromatic index, stating that the fractional f -chromatic index equals the maximum of the fractional maximum f -de...

An upper bound is given on the number of acyclic orientations of a graph, in terms of the spectrum of its Laplacian. It is shown that this improves upon the previously known bound, which depended on the degree sequence of the graph. Estimates on the new bound are provided. A lower bound on the number of acyclic orientations of a graph is given, with the help of the probabilistic method. This ar...

We prove, by means of explicit bijections, theorems of Whitney and Stanley that express the coefficients of the chromatic polynomial of a graph G and the number of acyclic orientations of G in terms of numbers of sets of edges that contain no broken circuits of G. Let G be a graph, and let a total ordering of its edge set E ( G ) be fixed. Words like “first” or “later,” when applied to edges, w...

A strong edge coloring of a graph G is an edge coloring such that every two adjacent edges or two edges adjacent to a same edge receive two distinct colors; in other words, every path of length three has three distinct colors in G. The strong chromatic index of G, denoted by   S G  , is the smallest integer k such that G admits a strong edge coloring with k colors. This survey is an brief i...

In [4], Cariolaro et al. demonstrated how colorability problems can be approached numerically by the use of computer algebra systems and the Combinatorial Nullstellensatz. In particular, they verified a case of the List Coloring Conjecture by proving that the list-chromatic index of K6 is 5. In this short note, we show that using the coefficient formula of Schauz [16] is much more efficient tha...

A proper vertex coloring ? of graph G is said to be odd if for each non-isolated x?V(G) there exists a color c such that ??1(c)?N(x) odd-sized. The minimum number colors in any G, denoted ?o(G), the chromatic number. Odd colorings were recently introduced Petruševski and Škrekovski (0000). Here we discuss various basic properties this new parameter, characterize acyclic graphs hypercubes terms ...