Two results on the digraph chromatic number

It is known (Bollob\'{a}s (1978); Kostochka and Mazurova (1977)) that there exist graphs of maximum degree $\Delta$ and of arbitrarily large girth whose chromatic number is at least $c \Delta / \log \Delta$. We show an analogous result for digraphs where the chromatic number of a digraph $D$ is defined as the minimum integer $k$ so that $V(D)$ can be partitioned into $k$ acyclic sets, and the girth is the length of the shortest cycle in the corresponding undirected graph. It is also shown, in the same vein as an old result of Erdos (1962), that there are digraphs with arbitrarily large chromatic number where every large subset of vertices is 2-colorable

Winding number and circular 4-coloring of signed graphs

Concerning the recent notion of circular chromatic number of signed graphs, for each given integer $k$ we introduce two signed bipartite graphs, each on $2k^2-k+1$ vertices, having shortest negative cycle of length $2k$, and the circular chromatic number 4. Each of the construction can be viewed as a bipartite analogue of the generalized Mycielski graphs on odd cycles, $M_{\ell}(C_{2k+1})$. In the course of proving our result, we also obtain a simple proof of the fact that $M_{\ell}(C_{2k+1})$ and some similar quadrangulations of the projective plane have circular chromatic number 4. These proofs have the advantage that they illuminate, in an elementary manner, the strong relation between algebraic topology and graph coloring problems.Comment: 16 pages, 11 figure

Small odd cycles in 4-chromatic graphs

It is shown that every 4-chromatic graph on n vertices contains an odd cycle of length less than 2√n+3. This improves the previous bound given by Nilli [J Graph Theory 3 (1999), 145-147]

Small Odd Cycles in 4-Chromatic Graphs

It is shown that every 4-chromatic graph on n vertices contains an odd cycle of length less than 2 p n ‡3. This improves the previous boun