Distance colouring without one cycle length

We consider distance colourings in graphs of maximum degree at most $d$ and how excluding one fixed cycle length $\ell$ affects the number of colours required as $d\to\infty$. For vertex-colouring and $t\ge 1$, if any two distinct vertices connected by a path of at most $t$ edges are required to be coloured differently, then a reduction by a logarithmic (in $d$) factor against the trivial bound $O(d^t)$ can be obtained by excluding an odd cycle length $\ell \ge 3t$ if $t$ is odd or by excluding an even cycle length $\ell \ge 2t+2$. For edge-colouring and $t\ge 2$, if any two distinct edges connected by a path of fewer than $t$ edges are required to be coloured differently, then excluding an even cycle length $\ell \ge 2t$ is sufficient for a logarithmic factor reduction. For $t\ge 2$, neither of the above statements are possible for other parity combinations of $\ell$ and $t$. These results can be considered extensions of results due to Johansson (1996) and Mahdian (2000), and are related to open problems of Alon and Mohar (2002) and Kaiser and Kang (2014).Comment: 14 pages, 1 figur

Maximum Δ\Delta-edge-colorable subgraphs of class II graphs

A graph $G$ is class II, if its chromatic index is at least $\Delta+1$. Let $H$ be a maximum $\Delta$-edge-colorable subgraph of $G$. The paper proves best possible lower bounds for $\frac{|E(H)|}{|E(G)|}$, and structural properties of maximum $\Delta$-edge-colorable subgraphs. It is shown that every set of vertex-disjoint cycles of a class II graph with $\Delta\geq3$ can be extended to a maximum $\Delta$-edge-colorable subgraph. Simple graphs have a maximum $\Delta$-edge-colorable subgraph such that the complement is a matching. Furthermore, a maximum $\Delta$-edge-colorable subgraph of a simple graph is always class I.Comment: 13 pages, 2 figures, the proof of the Lemma 1 is correcte

Perfect packings with complete graphs minus an edge

Let K_r^- denote the graph obtained from K_r by deleting one edge. We show that for every integer r\ge 4 there exists an integer n_0=n_0(r) such that every graph G whose order n\ge n_0 is divisible by r and whose minimum degree is at least (1-1/chi_{cr}(K_r^-))n contains a perfect K_r^- packing, i.e. a collection of disjoint copies of K_r^- which covers all vertices of G. Here chi_{cr}(K_r^-)=r(r-2)/(r-1) is the critical chromatic number of K_r^-. The bound on the minimum degree is best possible and confirms a conjecture of Kawarabayashi for large n

Acyclic edge coloring of graphs

An {\em acyclic edge coloring} of a graph $G$ is a proper edge coloring such that the subgraph induced by any two color classes is a linear forest (an acyclic graph with maximum degree at most two). The {\em acyclic chromatic index} \chiup_{a}'(G) of a graph $G$ is the least number of colors needed in an acyclic edge coloring of $G$. Fiam\v{c}\'{i}k (1978) conjectured that \chiup_{a}'(G) \leq \Delta(G) + 2, where $\Delta(G)$ is the maximum degree of $G$. This conjecture is well known as Acyclic Edge Coloring Conjecture (AECC). A graph $G$ with maximum degree at most $\kappa$ is {\em $\kappa$-deletion-minimal} if \chiup_{a}'(G) > \kappa and \chiup_{a}'(H) \leq \kappa for every proper subgraph $H$ of $G$. The purpose of this paper is to provide many structural lemmas on $\kappa$-deletion-minimal graphs. By using the structural lemmas, we firstly prove that AECC is true for the graphs with maximum average degree less than four (\autoref{NMAD4}). We secondly prove that AECC is true for the planar graphs without triangles adjacent to cycles of length at most four, with an additional condition that every $5$-cycle has at most three edges contained in triangles (\autoref{NoAdjacent}), from which we can conclude some known results as corollaries. We thirdly prove that every planar graph $G$ without intersecting triangles satisfies \chiup_{a}'(G) \leq \Delta(G) + 3 (\autoref{NoIntersect}). Finally, we consider one extreme case and prove it: if $G$ is a graph with $\Delta(G) \geq 3$ and all the $3^{+}$-vertices are independent, then \chiup_{a}'(G) = \Delta(G). We hope the structural lemmas will shed some light on the acyclic edge coloring problems.Comment: 19 page