Uniquely $D$-colourable digraphs with large girth II: simplification via generalization
P. Mark Kayll, Esmaeil Parsa
备注：
15 pages, 0 figures, minor revisions to address referee comments, revision in Section 4 (Case II) to expand and clarify proof, to appear in Electronic Journal of Combinatorics
We prove that for every digraph $D$ and every choice of positive integers $k$, $\ell$ there exists a digraph $D^*$ with girth at least $\ell$ together with a surjective acyclic homomorphism $ψ\colon D^*\to D$ such that: (i) for every digraph $C$ of order at most $k$, there exists an acyclic homomorphism $D^*\to C$ if and only if there exists an acyclic homomorphism $D\to C$; and (ii) for every $D$-pointed digraph $C$ of order at most $k$ and every acyclic homomorphism $φ\colon D^*\to C$ there exists a unique acyclic homomorphism $f\colon D\to C$ such that $φ=f\circψ$. This implies the main results in [A. Harutyunyan et al., Uniquely $D$-colourable digraphs with large girth, Canad. J. Math., 64(6) (2012), 1310-1328; MR2994666] analogously with how the work [J. Nešetřil and X. Zhu, On sparse graphs with given colorings and homomorphisms, J. Combin. Theory Ser. B, 90(1) (2004), 161-172; MR2041324] generalizes and extends [X. Zhu, Uniquely $H$-colorable graphs with large girth, J. Graph Theory, 23(1) (1996), 33-41; MR1402136].
Let C and D be digraphs. A mapping $f:V(D)\to V(C)$ is a C-colouring 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-colourable if it admits a C-colouring and that D is uniquely C-colourable if it is surjectively C-colourable and any two C-colourings of D differ by an automorphism of C. We prove that if a digraph D is not C-colourable, then there exist digraphs of arbitrarily large girth that are D-colourable but not C-colourable. Moreover, for every digraph D that is uniqu…
Neumann-Lara conjectured in 1985 that every planar digraph with digirth at least three is 2-colourable, meaning that the vertices can be 2-coloured without creating any monochromatic directed cycles. We prove a relaxed version of this conjecture: every planar digraph of digirth at least four is 2-colourable.
Let $D$ be a strongly connected digraph. An arc set $S$ of $D$ is a \emph{restricted arc-cut} of $D$ if $D-S$ has a non-trivial strong component $D_{1}$ such that $D-V(D_{1})$ contains an arc. The \emph{restricted arc-connectivity} $λ'(D)$ of a digraph $D$ is the minimum cardinality over all restricted arc-cuts of $D$. A strongly connected digraph $D$ is \emph{$λ'$-connected} when $λ'(D)$ exists. This paper presents a family $\cal{F}$ of strong digraphs of girth four that are not $λ'$-connected and for every strong digraph $D\notin \cal{F}$ with girth four it follows that it is $λ'$-connected.…
The digirth of a digraph is the length of a shortest directed cycle. The dichromatic number $\vecχ(D)$ of a digraph $D$ is the smallest size of a partition of the vertex-set into subsets inducing acyclic subgraphs. A conjecture by Harutyunyan and Mohar states that $\vecχ(D) \le \left\lceil\fracΔ{4}\right\rceil+1$ for every digraph $D$ of digirth at least $3$ and maximum degree $Δ$. The best known partial result by Golowich shows that $\vecχ(D) \le \frac{2}{5}Δ+O(1)$. In this short note we prove for every $g \ge 2$ that if $D$ is a digraph of digirth at least $2g-1$ and maximum degree $Δ$, then…
Neumann-Lara and Škrekovski conjectured that every planar digraph is $2$-colourable. We show that this conjecture is equivalent to the more general statement that all oriented $K_5$-minor-free graphs are $2$-colourable.
Suzuki (2004) [7] classified thin weakly distance-regular digraphs and pro- posed the project to classify weakly distance-regular digraphs of valency 3. The case of girth 2 was classified by the third author (2004) [9] under the assumption of the commutativity. In this paper, we continue this project and classify these digraphs with girth more than 2 and two types of arcs.
Brooks' Theorem states that a connected graph $G$ of maximum degree $Δ$ has chromatic number at most $Δ$, unless $G$ is an odd cycle or a complete graph. A result of Johansson (1996) shows that if $G$ is triangle-free, then the chromatic number drops to $O(Δ/ \log Δ)$. In this paper, we derive a weak analog for the chromatic number of digraphs. We show that every (loopless) digraph $D$ without directed cycles of length two has chromatic number $χ(D) \leq (1-e^{-13}) \tildeΔ$, where $\tildeΔ$ is the maximum geometric mean of the out-degree and in-degree of a vertex in $D$, when $\tildeΔ$ is suf…
We consider acyclic r-colorings in graphs and digraphs: they color the vertices in r colors, each of which induces an acyclic graph or digraph. (This includes the dichromatic number of a digraph, and the arboricity of a graph.) For any girth and sufficiently high degree, we prove the NP-completeness of acyclic r-colorings; our method also implies the known analogue for classical colorings. The proofs use high girth graphs with high arboricity and dichromatic numbers. High girth graphs and digraphs with high chromatic and dichromatic numbers have been well studied; we re-derive the results from…
Let $k$ be a positive integer. Bermond and Thomassen conjectured in 1981 that every digraph with minimum outdegree at least $2k-1$ contains $k$ vertex-disjoint cycles. It is famous as one of the one hundred unsolved problems selected in [Bondy, Murty, Graph Theory, Springer-Verlag London, 2008]. Lichiardopol, Por and Sereni proved in [SIAM J. Discrete Math. 23 (2) (2009) 979-992] that the above conjecture holds for $k=3$. Let $g$ be the girth, i.e., the length of the shortest cycle, of a given digraph. Bang-Jensen, Bessy and Thomassé conjectured in [J. Graph Theory 75 (3) (2014) 284-302] that …
In the theory of digraphs, the study of cycles is a subject of great importance and has given birth to a number of deep questions such as the Behzad-Chartrand-Wall conjecture (1970) and its generalization, the Caccetta-Häggkvist conjecture (1978). Despite a lot of interest and efforts, the progress on these remains slow and mostly restricted to the solution of some special cases. In this note, we prove these conjectures for digraphs with girth is at least as large as their minimum out-degree and without short even cycles. More generally, we prove that if a digraph has sufficiently large girth …
It is known (Bollobás (1978); Kostochka and Mazurova (1977)) that there exist graphs of maximum degree $Δ$ and of arbitrarily large girth whose chromatic number is at least $c Δ/ \log Δ$. 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 ever…
In this paper, we classify all commutative weakly distance-regular digraphs of girth $g$ and one type of arcs under the assumption that $p_{(1,g-1),(1,g-1)}^{(2,g-2)}\geq k_{1,g-1}-2$. In consequence, we recover [13, Theorem 1.1] as a special case of our result.
For a fixed graph H, the H-Recoloring problem asks whether for two given homomorphisms from a graph G to H, we can transform one into the other by changing the image of a single vertex of G in each step and maintaining a homomorphism from G to H throughout. We extend an algorithm of Wrochna for H-Recoloring where H is a square-free loopless undirected graph to the more general setting of directed graphs. We obtain a polynomial-time algorithm for H-Recoloring in this setting whenever H is a loopless digraph that does not contain a 4-cycle of algebraic girth zero and whenever H is a reflexive di…
In 1959, Grötzsch famously proved that every planar graph of girth at least 4 is 3-colourable (or equivalently, admits a homomorphism to $C_3$). A natural generalization of this is the following conjecture: for every positive integer $t$, every planar graph of girth at least $4t$ admits a homomorphism to $C_{2t+1}$. This is in fact the planar dual of a well-known conjecture of Jaeger which states that every $4t$-edge-connected graph admits a modulo $(2t+1)$-orientation. Though Jaeger's original conjecture was disproved in 2018 by Han et al., Lovasz et al. showed that every $6t$-edge connected …
It is known that every distance-regular digraph is connected and normal. An interesting question is: when is a given connected normal digraph distance-regular? Motivated by this question first we give some characterizations of weakly distance-regular digraphs. Specially we show that whether a given connected digraph to be weakly distance-regular only depends on the equality for two invariants. Then we show that a connected normal digraph $\G$ with $d+1$ distinct eigenvalues is distance-regular if and only if the simple excess (the ratio of the square of mean of the numbers of shortest paths be…
In the homomorphism order of digraphs, a duality pair is an ordered pair of digraphs $(G,H)$ such that for any digraph, $D$, $G\to D$ if and only if $D\not\to H$. The directed path on $k+1$ vertices together with the transitive tournament on $k$ vertices is a classic example of a duality pair. This relation between paths and tournaments implies that a graph is $k$-colourable if and only if it admits an orientation with no directed path on more than $k$-vertices. In this work, for every undirected cycle $C$ we find an orientation $C_D$ and an oriented path $P_C$, such that $(P_C,C_D)$ is a dual…
