Uniquely D-colourable digraphs with large girth

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 uniquely D-colourable, there exists a uniquely D-colourable digraph of arbitrarily large girth. In particular, this implies that for every rational number $r\geq 1$, there are uniquely circularly r-colourable digraphs with arbitrarily large girth.Comment: 21 pages, 0 figures To be published in Canadian Journal of Mathematic

Hole의 관점에서 그래프와 유향그래프의 구조에 관한 연구

학위논문(박사)--서울대학교 대학원 :사범대학 수학교육과,2019. 8. 김서령.이 논문에서는 유향그래프와 그래프의 홀의 관점에서 계통발생 그래프와 그래프의 삼각화에 대하여 연구한다. 길이 4 이상인 유도된 싸이클을 홀이라 하고 홀이 없는 그래프를 삼각화된 그래프라 한다. 구체적으로, 싸이클을 갖지 않는 유향그래프의 계통발생 그래프가 삼각화된 그래프인지 판정하고, 주어진 그래프를 삼각화하여 클릭수가 크게 차이 나지 않는 그래프를 만드는 방법을 찾고자 한다. 이 논문은 연구 내용에 따라 두 부분으로 나뉜다. 먼저 $(1, i)$ 유향그래프와 $(i, 1)$ 유향그래프의 계통발생 그래프를 완전하게 특징화하고, $(2, j)$ 유향그래프 $D$의 모든 유향변에서 방향을 제거한 그래프가 삼각화된 그래프이면, $D$의 계통발생 그래프 역시 삼각화된 그래프임을 보였다. 또한 적은 수의 삼각형을 갖는 연결된 그래프의 계통발생수를 계산한 정리를 확장하여 많은 수의 삼각형을 포함한 연결된 그래프의 계통발생수를 계산하였다. 다른 한 편 그래프 $G$의 비삼각화 지수 $i(G)$에 대하여 $\omega(G^*)-\omega(G) \le i(G)$를 만족하는 $G$의 삼각화된 그래프 $G^*$가 존재함을 보였다. 그리고 이를 도구로 이용하여 NC property를 만족하는 그래프가 Hadwiger 추측과 Erd\H{o}s-Faber-Lov\'{a}sz 추측을 만족함을 증명하고, 비삼각화 지수가 유계인 그래프들이 linearly $\chi$-bounded임을 증명하였다.This thesis aims at studying phylogeny graphs and graph completions in the aspect of holes of graphs or digraphs. A hole of a graph is an induced cycle of length at least four and a graph is chordal if it does not contain a hole. Specifically, we determine whether the phylogeny graphs of acyclic digraphs are chordal or not and find a way of chordalizing a graph without increasing the size of maximum clique not so much. In this vein, the thesis is divided into two parts. In the first part, we completely characterize phylogeny graphs of $(1, i)$ digraphs and $(i,1)$ digraphs, respectively, for a positive integer $i$. Then, we show that the phylogeny graph of a $(2,j)$ digraph $D$ is chordal if the underlying graph of $D$ is chordal for any positive integer $j$. In addition, we extend the existing theorems computing phylogeny numbers of connected graph with a small number of triangles to results computing phylogeny numbers of connected graphs with many triangles. In the second part, we present a minimal chordal supergraph $G^*$ of a graph $G$ satisfying the inequality $\omega(G^*) - \omega(G) \le i(G)$ for the non-chordality index $i(G)$ of $G$. Using the above chordal supergraph as a tool, we prove that the family of graphs satisfying the NC property satisfies the Hadwiger conjecture and the Erd\H{o}s-Faber-Lov\'{a}sz Conjecture, and the family of graphs with bounded non-chordality indices is linearly $\chi$-bounded.Contents Abstract i 1 Introduction 1 1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Phylogeny graphs . . . . . . . . . . . . . . . . . . . . . 8 1.2.2 Graph colorings and chordal completions . . . . . . . . 14 2 Phylogeny graphs 19 2.1 Chordal phylogeny graphs . . . . . . . . . . . . . . . . . . . . 19 2.1.1 (1,j) phylogeny graphs and (i,1) phylogeny graphs . . 20 2.1.2 (2,j) phylogeny graphs . . . . . . . . . . . . . . . . . . 28 2.2 The phylogeny number and the triangles and the diamonds of a graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3 A new minimal chordal completion 61 3.1 Graphs with the NC property . . . . . . . . . . . . . . . . . . 64 3.2 The Erd˝ os-Faber-Lovász Conjecture . . . . . . . . . . . . . . . 73 3.3 A minimal chordal completion of a graph . . . . . . . . . . . . 80 3.3.1 Non-chordality indices of graphs . . . . . . . . . . . . . 80 3.3.2 Making a local chordalization really local . . . . . . . . 89 3.4 New χ-bounded classes . . . . . . . . . . . . . . . . . . . . . . 97 Abstract (in Korean) 107Docto

Directed acyclic graphs with the unique dipath property

International audienceLet P be a family of dipaths of a DAG (Directed Acyclic Graph) G. The load of an arc is the number of dipaths containing this arc. Let π(G, P) be the maximum of the load of all the arcs and let w(G, P) be the minimum number of wavelengths (colors) needed to color the family of dipaths P in such a way that two dipaths with the same wavelength are arc-disjoint. There exist DAGs such that the ratio between w(G, P) and π(G, P) cannot be bounded. An internal cycle is an oriented cycle such that all the vertices have at least one predecessor and one successor in G (said otherwise every cycle contain neither a source nor a sink of G). We prove that, for any family of dipaths P, w(G, P) = π(G, P) if and only if G is without internal cycle. We also consider a new class of DAGs, which is of interest in itself, those for which there is at most one dipath from a vertex to another. We call these digraphs UPP-DAGs. For these UPP-DAGs we show that the load is equal to the maximum size of a clique of the conflict graph. We prove that the ratio between w(G, P) and π(G, P) cannot be bounded (a result conjectured in an other article). For that we introduce "good labelings" of the conflict graph associated to G and P, namely labelings of the edges such that for any ordered pair of vertices (x, y) there do not exist two paths from x to y with increasing labels

Proximity Drawings of High-Degree Trees

A drawing of a given (abstract) tree that is a minimum spanning tree of the vertex set is considered aesthetically pleasing. However, such a drawing can only exist if the tree has maximum degree at most 6. What can be said for trees of higher degree? We approach this question by supposing that a partition or covering of the tree by subtrees of bounded degree is given. Then we show that if the partition or covering satisfies some natural properties, then there is a drawing of the entire tree such that each of the given subtrees is drawn as a minimum spanning tree of its vertex set

Directed acyclic graphs with the unique dipath property

Let PP be a family of dipaths of a DAG (Directed Acyclic Graph) G. The load of an arc is the number of dipaths containing this arc. Let pi (G,PP) be the maximum of the load of all the arcs and let w(G, PP) be the minimum number of wavelengths (colors) needed to color the family of dipaths PP in such a way that two dipaths with the same wavelength are arc-disjoint. There exist DAGs such that the ratio between w(G, PP) and pi (G,PP) cannot be bounded. An internal cycle is an oriented cycle such that all the vertices have at least one predecessor and one successor in G (said otherwise every cycle contains neither a source nor a sink of G). We prove that, for any family of dipaths PP, w(G, PP = pi(G,PP) if and only if G is without internal cycle. We also consider a new class of DAGs, which is of interest in itself, those for which there is at most one dipath from a vertex to another. We call these digraphs UPP-DAGs. For these UPP-DAGs we show that the load is equal to the maximum size of a clique of the conflict graph. We prove that the ratio between w(G, PP) and pi (G,PP) cannot be bounded (a result conjectured in an other article). For that we introduce ''good labelings'' of the conflict graph associated to G and PP, namely labelings of the edges such that for any ordered pair of vertices (x,y) there do not exist two paths from $x$ to $y$ with increasing labels

Degrees in oriented hypergraphs and Ramsey p-chromatic number

The family D(k,m) of graphs having an orientation such that for every vertex v ∈ V (G) either (outdegree) deg+(v) ≤ k or (indegree) deg−(v) ≤ m have been investigated recently in several papers because of the role D(k,m) plays in the efforts to estimate the maximum directed cut in digraphs and the minimum cover of digraphs by directed cuts. Results concerning the chromatic number of graphs in the family D(k,m) have been obtained via the notion of d-degeneracy of graphs. In this paper we consider a far reaching generalization of the family D(k,m), in a complementary form, into the context of r-uniform hypergraphs, using a generalization of Hakimi’s theorem to r-uniform hypergraphs and by showingPeer ReviewedPostprint (published version