Conjectures on uniquely 3-edge-colorable graphs

A graph $G$ is {\it uniquely k-edge-colorable} if the chromatic index of $G$ is $k$ and every two $k$-edge-colorings of $G$ produce the same partition of $E(G)$ into $k$ independent subsets.For any $k\ne 3$, a uniquely $k$-edge-colorable graph $G$ is completely characterized;$G\cong K_2$ if $k=1$, $G$ is a path or an even cycle if $k=2$,and $G$ is a star $K_{1,k}$ if $k\geq 4$.On the other hand, there are infinitely many uniquely 3-edge-colorable graphs, and hence, there are many conjectures for the characterization of uniquely 3-edge-colorable graphs.In this paper, we introduce a new conjecture which connects conjectures of uniquely 3-edge-colorable planar graphs with those of uniquely 3-edge-colorable non-planar graphs

Uniquely Distinguishing Colorable Graphs

A graph is called uniquely distinguishing colorable if there is only one partition of vertices of the graph that forms distinguishing coloring with the smallest possible colors. In this paper, we study the unique colorability of the distinguishing coloring of a graph and its applications in computing the distinguishing chromatic number of disconnected graphs. We introduce two families of uniquely distinguishing colorable graphs, namely type 1 and type 2, and show that every disconnected uniquely distinguishing colorable graph is the union of two isomorphic graphs of type 2. We obtain some results on bipartite uniquely distinguishing colorable graphs and show that any uniquely distinguishing $n$-colorable tree with $n \geq 3$ is a star graph. For a connected graph $G$, we prove that $\chi_D(G\cup G)=\chi_D(G)+1$ if and only if $G$ is uniquely distinguishing colorable of type 1. Also, a characterization of all graphs $G$ of order $n$ with the property that $\chi_{D}(G\cup G) = \chi_{D}(G) = k$, where $k=n-2, n-1, n$, is given in this paper. Moreover, we determine all graphs $G$ of order $n$ with the property that $\chi_{D}(G\cup G) = \chi_{D}(G)+1 = \ell$, where $\ell=n-1, n, n+1$. Finally, we investigate the family of connected graphs $G$ with $\chi_{D}(G\cup G) = \chi_{D}(G)+1 = 3$

Algebraic Characterization of Uniquely Vertex Colorable Graphs

The study of graph vertex colorability from an algebraic perspective has introduced novel techniques and algorithms into the field. For instance, it is known that $k$-colorability of a graph $G$ is equivalent to the condition $1 \in I_{G,k}$ for a certain ideal I_{G,k} \subseteq \k[x_1, ..., x_n]. In this paper, we extend this result by proving a general decomposition theorem for $I_{G,k}$. This theorem allows us to give an algebraic characterization of uniquely $k$-colorable graphs. Our results also give algorithms for testing unique colorability. As an application, we verify a counterexample to a conjecture of Xu concerning uniquely 3-colorable graphs without triangles.Comment: 15 pages, 2 figures, print version, to appear J. Comb. Th. Ser.

D-colorable digraphs with large girth

In 1959 Paul Erdos (Graph theory and probability, Canad. J. Math. 11 (1959), 34-38) famously proved, nonconstructively, that there exist graphs that have both arbitrarily large girth and arbitrarily large chromatic number. This result, along with its proof, has had a number of descendants (D. Bokal, G. Fijavz, M. Juvan, P.M. Kayll and B. Mohar, The circular chromatic number of a digraph, J. Graph Theory 46 (2004), 227-240; B. Bollobas and N. Sauer, Uniquely colourable graphs with large girth, Canad. J. Math. 28 (1976), 1340-1344; J. Nesetril and X. Zhu, On sparse graphs with given colorings and homomorphisms, J. Combin. Theory Ser. B 90 (2004), 161-172; X. Zhu, Uniquely H-colorable graphs with large girth, J. Graph Theory 23 (1996), 33-41) that have extended and generalized the result while strengthening the techniques used to achieve it. We follow the lead of Xuding Zhu (op. cit.) who proved that, for a suitable graph H, there exist graphs of arbitrarily large girth that are uniquely H-colorable. We establish an analogue of Zhu\u27s results in a digraph setting. Let C and D be digraphs. A mapping f:V(D)&rarr V(C) is a C-coloring if for every arc uv of D, either f(u)f(v) is an arc of C or f(u)=f(v), and the preimage of every vertex of C induces an acyclic subdigraph in D. We say that D is C-colorable if it admits a C-coloring and that D is uniquely C-colorable if it is surjectively C-colorable and any two C-colorings of D differ by an automorphism of C. We prove that if D is a digraph that is not C-colorable, then there exist graphs of arbitrarily large girth that are D-colorable but not C-colorable. Moreover, for every digraph D that is uniquely D-colorable, there exists a uniquely D-colorable digraph of arbitrarily large girth