Edge-choosability of multicircuits

AbstractA multicircuit is a multigraph whose underlying simple graph is a circuit (a connected 2-regular graph). The List-Colouring Conjecture (LCC) is that every multigraph G has edge-choosability (list chromatic index) ch′(G) equal to its chromatic index χ′(G). In this paper the LCC is proved first for multicircuits, and then, building on results of Peterson and Woodall, for any multigraph G in which every block is bipartite or a multicircuit or has at most four vertices or has underlying simple graph of the form K1, 1, p

Star 5-edge-colorings of subcubic multigraphs

The star chromatic index of a multigraph $G$, denoted $\chi'_{s}(G)$, is the minimum number of colors needed to properly color the edges of $G$ such that no path or cycle of length four is bi-colored. A multigraph $G$ is star $k$-edge-colorable if $\chi'_{s}(G)\le k$. Dvo\v{r}\'ak, Mohar and \v{S}\'amal [Star chromatic index, J Graph Theory 72 (2013), 313--326] proved that every subcubic multigraph is star $7$-edge-colorable, and conjectured that every subcubic multigraph should be star $6$-edge-colorable. Kerdjoudj, Kostochka and Raspaud considered the list version of this problem for simple graphs and proved that every subcubic graph with maximum average degree less than $7/3$ is star list-$5$-edge-colorable. It is known that a graph with maximum average degree $14/5$ is not necessarily star $5$-edge-colorable. In this paper, we prove that every subcubic multigraph with maximum average degree less than $12/5$ is star $5$-edge-colorable.Comment: to appear in Discrete Mathematics. arXiv admin note: text overlap with arXiv:1701.0410

A note on total and list edge-colouring of graphs of tree-width 3

It is shown that Halin graphs are $\Delta$-edge-choosable and that graphs of tree-width 3 are $(\Delta+1)$-edge-choosable and $(\Delta +2)$-total-colourable.Comment: arXiv admin note: substantial text overlap with arXiv:1504.0212

Bounded degree graphs and hypergraphs with no full rainbow matchings

Given a multi-hypergraph $G$ that is edge-colored into color classes $E_1, \ldots, E_n$, a full rainbow matching is a matching of $G$ that contains exactly one edge from each color class $E_i$. One way to guarantee the existence of a full rainbow matching is to have the size of each color class $E_i$ be sufficiently large compared to the maximum degree of $G$. In this paper, we apply a simple iterative method to construct edge-colored multi-hypergraphs with a given maximum degree, large color classes, and no full rainbow matchings. First, for every $r \ge 1$ and $\Delta \ge 2$, we construct edge-colored $r$-uniform multi-hypergraphs with maximum degree $\Delta$ such that each color class has size $|E_i| \ge r\Delta - 1$ and there is no full rainbow matching, which demonstrates that a theorem of Aharoni, Berger, and Meshulam (2005) is best possible. Second, we construct properly edge-colored multigraphs with no full rainbow matchings which disprove conjectures of Delcourt and Postle (2022). Finally, we apply results on full rainbow matchings to list edge-colorings and prove that a color degree generalization of Galvin's theorem (1995) does not hold

Every Elementary Graph is Chromatic Choosable

Elementary graphs are graphs whose edges can be colored using two colors in such a way that the edges in any induced $P_3$ get distinct colors. They constitute a subclass of the class of claw-free perfect graphs. In this paper, we show that for any elementary graph, its list chromatic number and chromatic number are equal