On topological relaxations of chromatic conjectures

There are several famous unsolved conjectures about the chromatic number that were relaxed and already proven to hold for the fractional chromatic number. We discuss similar relaxations for the topological lower bound(s) of the chromatic number. In particular, we prove that such a relaxed version is true for the Behzad-Vizing conjecture and also discuss the conjectures of Hedetniemi and of Hadwiger from this point of view. For the latter, a similar statement was already proven in an earlier paper of the first author with G. Tardos, our main concern here is that the so-called odd Hadwiger conjecture looks much more difficult in this respect. We prove that the statement of the odd Hadwiger conjecture holds for large enough Kneser graphs and Schrijver graphs of any fixed chromatic number

On The b-Chromatic Number of Regular Graphs Without 4-Cycle

The b-chromatic number of a graph $G$, denoted by $\phi(G)$, is the largest integer $k$ that $G$ admits a proper $k$-coloring such that each color class has a vertex that is adjacent to at least one vertex in each of the other color classes. We prove that for each $d$-regular graph $G$ which contains no 4-cycle, $\phi(G)\geq\lfloor\frac{d+3}{2}\rfloor$ and if $G$ has a triangle, then $\phi(G)\geq\lfloor\frac{d+4}{2}\rfloor$. Also, if $G$ is a $d$-regular graph which contains no 4-cycle and $diam(G)\geq6$, then $\phi(G)=d+1$. Finally, we show that for any $d$-regular graph $G$ which does not contain 4-cycle and $\kappa(G)\leq\frac{d+1}{2}$, $\phi(G)=d+1$

On game chromatic number analogues of Mycielsians and Brooks' Theorem

The vertex coloring game is a two-player game on a graph with given color set in which the first player attempts to properly color the graph and the second attempts to prevent a proper coloring from being achieved. The smallest number of colors for which the first player can win no matter how the second player plays is called the game chromatic number of the graph. In this paper we initiate the study of game chromatic number for Mycielskians and a game chromatic number analogue of Brooks' Theorem (which characterizes graphs for which chromatic number is at most the maximum degree of the graph). In particular, we determine the game chromatic number of Mycielskians of complete graphs, complete bipartite graphs, and cycles. In the direction of Brooks' Theorem, we show that if there are few vertices of maximum degree or if all vertices of maximum degree are at least three edges apart, then the game chromatic number is at most the maximum degree of the grap