Hongliang Lu, Bing Bai, Wei Wang
Let $k$, $m$ and $r$ be three integers such that $2\leq k\leq m\leq r$. Let $G$ be a $2r$-regular, $2m$-edge-connected graph of odd order. We obtain some sufficient conditions, which guarantee $G-v$ contains a $k$-factor for all $v\in V(G)$.
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Understanding the structure of a graph along with the structure of its subgraphs is important for several problems in graph theory. Two examples are the Reconstruction Conjecture and isomorph-free generation. This paper raises the question of which pairs of groups can be represented as the automorphism groups of a graph and a vertex-deleted subgraph. This, and more surprisingly the analogous question for edge-deleted subgraphs, are answered in the most positive sense using concrete constructions.
A b-coloring of a graph is a proper coloring of its vertices such that each color class contains a vertex adjacent to at least one vertex of every other color class. The b-chromatic number of a graph is the largest integer k such that the graph has a b-coloring with k colors. In this work we present lower bounds for the b-chromatic number of a vertex-deleted subgraph of a graph, particularly regarding two important classes, quasi-line and chordal graphs. We also get bounds for the b-chromatic number of G -{x}, when G is a graph with large girth.
Given a $k$-vertex graph $H$ and an integer $n$, what are the $n$-vertex graphs with the maximum number of induced copies of $H$? This question is closely related to the inducibility problem introduced by Pippenger and Golumbic in 1975, which asks for the maximum possible fraction of $k$-vertex subsets of an $n$-vertex graph that induce a copy of $H$. Huang, Lee and the first author proved that for a random $k$-vertex graph $H$, almost surely the $n$-vertex graphs maximizing the number of induced copies of $H$ are the balanced iterated blow-ups of $H$. In this paper, we consider the case where…
A characterization of generalized cospectrality of rooted graphs with applications in graph reconstruction
本文通过矩阵刻画具有广义同谱删顶点子图的根图，并据此给出图可重构性的新条件，证明含几乎可控且具非平凡自同构删点子图的图可重构。
The paper shows that almost every $n$-vertex graph is such that the multiset of its induced subgraphs on $3 \log_2{n}$ vertices is sufficient to determine it up to isomorphism. Therefore, for checking the isomorphism of a pair of $n$-vertex graphs, almost surely the multiset of their $3 \log_2{n}$-vertex subgraphs is sufficient .
Any finite group can be encoded as the automorphism group of an unlabeled simple graph. Recently Hartke, Kolb, Nishikawa, and Stolee (2010) demonstrated a construction that allows any ordered pair of finite groups to be represented as the automorphism group of a graph and a vertex-deleted subgraph. In this note, we describe a generalized scenario as a game between a player and an adversary: An adversary provides a list of finite groups and a number of rounds. The player constructs a graph with automorphism group isomorphic to the first group. In the following rounds, the adversary selects a gr…
Reconstructing edge-deleted unicyclic graphs
本文证明Harary重构猜想对恰好含一个环且具有三个非同构子树的图成立，改进了Müller和Lovás的界限。
This paper investigates vertex colorings of graphs such that some rainbow subgraph~$R$ and some monochromatic subgraph $M$ are forbidden. Previous work focussed on the case that $R=M$. Here we consider the more general case, especially the case that $M=K_2$.
Nonisomorphic Graphs Can Share an Arbitrarily Large Fraction of Their Vertex-Deleted Cards
该研究突破BBF预测，构造78顶点非同构图对有51张公共卡片，还对任意偶数r≥4构造图族使公共卡片比例渐近≥1-1/r，证明非同构图可共享任意大比例顶点删除卡片。
Suns in triangle-free graphs of large chromatic number
研究三角自由图中色数较大时是否存在诱导子图为特定太阳结构，证明色数至少为48的图存在特定太阳结构或近似结构。
Discrete Quantum Walks with Marked Vertices and Their Average Vertex Mixing Matrices
本文研究了正则图上带有标记顶点的离散量子游走，推导了平均顶点混合矩阵的公式与界限，并证明了界限紧致的条件及矩阵性质。
For a given graph $H$ we define $ρ(H)$ to be the minimum order of a graph $G$ such that every proper vertex coloring of $G$ contains a rainbow induced subgraph isomorphic to $H$. We give upper and lower bounds for $ρ(H)$, compute the exact value for some classes of graphs, and consider an interesting combinatorial problem connected with computation of $ρ(H)$ for paths. This research is motivated by some ideas in on-line graph coloring algorithms.
A refinement of Kelly's lemma for graph reconstruction for counting rooted subgraphs
本文精细化Kelly引理，研究计数根顶点与删除顶点重合的有根子图，证明一般不可能，但半径大于k时可计数固定高度k的多重集，并推广至边重构。
On vertex Ramsey graphs with forbidden subgraphs
本研究针对顶点Ramsey图的存在性问题，证明Nešetřil与Rödl提出的含禁止子图的顶点r-Ramsey图存在的充分条件，在颜色数r足够大时也是必要的，并将结果推广到图族及稠密二项随机图的典型存在性。
Let $G$ be an induced subgraph of the hypercube $Q_k$ for some $k$. We show that if $|G|$ is a power of $2$ then, for sufficiciently large $n$, the vertex set of $Q_n$ can be partitioned into induced copies of $G$. This answers a question of Offner. In fact, we prove a stronger statement: if $X$ is a subset of $\{0,1\}^k$ for some $k$ and if $|X|$ is a power of $2$, then, for sufficiently large $n$, $\{0,1\}^n$ can be partitioned into isometric copies of $X$.
If r(s, t) = p, we shall call Kp a Ramsey Minimal Example. We prove that if r{s,t) = p, and one edge is deleted from the Ramsey Minimal Example, Kp, the resulting graph no longer has the Ramsey property, that is, its edges can be colored red or blue so that the "red" subgraph does not contain Ks and the "blue" subgraph does not contain Kt. The paper is written in an accessible style explaining the process that led us to make the conjecture and then prove this result.
For a fixed graph $H$ on $k$ vertices, and a graph $G$ on at least $k$ vertices, we write $G\rightarrow H$ if in any vertex-coloring of $G$ with $k$ colors, there is an induced subgraph isomorphic to $H$ whose vertices have distinct colors. In other words, if $G\rightarrow H$ then a totally multicolored induced copy of $H$ is unavoidable in any vertex-coloring of $G$ with $k$ colors. In this paper, we show that, with a few notable exceptions, for any graph $H$ on $k$ vertices and for any graph $G$ which is not isomorphic to $H$, $G\not\!\rightarrow H$. We explicitly describe all exceptional ca…
A graph is called a chain graph if it is bipartite and the neighborhoods of the vertices in each color class form a chain with respect to inclusion. Alazemi, Andelić and Simić conjectured that no chain graph shares a non-zero (adjacency) eigenvalue with its vertex-deleted subgraphs. We disprove this conjecture. However, we show that the assertion holds for subgraphs obtained by deleting vertices of maximum degrees in either of color classes. We also give a simple proof for the fact that chain graphs have no eigenvalue in the interval $(0,1/2)$.
A $t$-bar visibility representation of a graph assigns each vertex up to $t$ horizontal bars in the plane so that two vertices are adjacent if and only if some bar for one vertex can see some bar for the other via an unobstructed vertical channel of positive width. The least $t$ such that $G$ has a $t$-bar visibility representation is the bar visibility number of $G$, denoted by $b(G)$. We show that if $H$ is a spanning subgraph of $G$, then $b(H)\le b(G)+1$. It follows that $b(G)\le \lceil n/6\rceil+1$ when $G$ is an $n$-vertex graph. This improves the upper bound obtained by Chang et al. (SI…
A note on vertex-critical induced subgraphs of shift graphs
本文研究了移位图中顶点临界诱导子图的性质，发现每个k色移位图包含唯一的k-顶点临界子图，并给出了其显式描述。
Many degree sequences can only be realised in graphs that contain a `ds-completable card', defined as a vertex-deleted subgraph in which the erstwhile neighbours of the deleted vertex can be identified from their degrees, if one knows the degree sequence of the original graph. We obtain conditions on the degree sequence, such that graphs whose degree sequence satisfies one of the conditions must contain such a card. The methods allow all such sequences on graphs of order up to 10 to be identified, and some fraction of the sequences for larger graphs. Among other applications, this can be used …
A path in a vertex-colored graph is called \emph{vertex-rainbow} if its internal vertices have pairwise distinct colors. A graph $G$ is \emph{rainbow vertex-connected} if for any two distinct vertices of $G$, there is a vertex-rainbow path connecting them. For a connected graph $G$, the \emph{rainbow vertex-connection number} of $G$, denoted by $rvc(G)$, is defined as the minimum number of colors that are required to make $G$ rainbow vertex-connected. In this paper, we find all the families $\mathcal{F}$ of connected graphs with $|\mathcal{F}|\in\{1,2\}$, for which there is a constant $k_\math…
Tree-independence number and forbidden induced subgraphs: excluding a $6$-vertex path and a $(2,t)$-biclique
研究排除6顶点路径和(2,t)-双 clique的图的树独立数上限，推进Dallard等人猜想的进展。
Note on extremal problems about connected subgraph sums
本文研究了图的连通子图和集合，以强形式证明了存在顶点赋值使得任意非同构图具有不同的连通子图和集合，并讨论了使所有连通子图和互异的顶点赋值。
We prove that for every integer $r\geq 2$, an $n$-vertex $k$-uniform hypergraph $H$ containing no $r$-regular subgraphs has at most $(1+o(1)){{n-1}\choose{k-1}}$ edges if $k\geq r+1$ and $n$ is sufficiently large. Moreover, if $r\in\{3,4\}$, $r\mid k$ and $k,n$ are both sufficiently large, then the maximum number of edges in an $n$-vertex $k$-uniform hypergraph containing no $r$-regular subgraphs is exactly ${{n-1} \choose {k-1}}$, with equality only if all edges contain a specific vertex $v$. We also ask some related questions.
Induced subgraph density. VII. The five-vertex path
本文证明了关于五顶点路径$P_5$的Erdős-Hajnal猜想，表明无诱导$P_5$的图中存在大规模的 clique 或 stable set。
A graph $G$ is $d$-degenerate if every non-null subgraph of $G$ has a vertex of degree at most $d$. We prove that every $n$-vertex planar graph has a $3$-degenerate induced subgraph of order at least $3n/4$.
For a graph $G$, we denote by $σ_{2}(G)$ the minimum degree sum of two non-adjacent vertices if $G$ is non-complete; otherwise, $σ_{2}(G) = +\infty$. In this paper, we prove the following two results: (i) If $s_{1}, s_{2} \ge 2$ are integers and $G$ is a non-complete graph with $σ_{2}(G) \ge 2(s_{1} + s_{2} + 1) - 1$, then $G$ contains two vertex-disjoint subgraphs $H_{1}$ and $H_{2}$ such that each $H_{i}$ is a graph of order at least $s_{i}+1$ with $σ_{2}(H_{i}) \ge 2s_{i} - 1$. (ii) If $s_{1}, s_{2} \ge 2$ are integers and $G$ is a triangle-free graph of order at least $3$ with $σ_{2}(G) \g…
Schrijver graphs are vertex-color-critical subgraphs of Kneser graphs having the same chromatic number. They also share the value of their fractional chromatic number but Schrijver graphs are not critical for that. Here we present an induced subgraph of every Schrijver graph that is vertex-critical with respect to the fractional chromatic number. These subgraphs turn out to be isomorphic with certain circular complete graphs. We also characterize the critical edges within this subgraph.
Inspired by the study of community structure in connection networks, we introduce the graph polynomial $Q(G;x,y)$, the bivariate generating function which counts the number of connected components in induced subgraphs. We give a recursive definition of $Q(G;x,y)$ using vertex deletion, vertex contraction and deletion of a vertex together with its neighborhood and prove a universality property. We relate $Q(G;x,y)$ to other known graph invariants and graph polynomials, among them partition functions, the Tutte polynomial, the independence and matching polynomials, and the universal edge elimina…
Sufficient conditions for fractional [a,b]-deleted graphs
本文研究分数 [a,b]-可删图的充分条件，通过建立图的边数、谱半径及无符号拉普拉斯谱半径的下界，保证图具有分数 [a,b]-可删性质。
On graphs without four-vertex induced subgraphs
本文针对著名的(claw, 4K1)-free图着色复杂度问题，通过建立结构定理，设计出多项式时间算法解决了其子类(claw, 4K1, bridge, C4-twin)-free图的着色问题。
Induced subgraphs and tree decompositions XI. Local structure in even-hole-free graphs of large treewidth
本文证明Sintiari-Trotignon猜想：大树宽无偶环图必含四顶点五边诱导子图，并推广到更一般的弦图与森林情形，且证明选择最优。
Given graphs H_1,...,H_k, we study the minimum order of a graph G such that for each i, the induced copies of H_i in G cover V(G). We prove a general upper bound of twice the sum of the numbers m_i, where m_i is one less than the order of H_i. When k=2 and one graph is an independent set of size n, we determine the optimum within a constant. When k=2 and the graphs are a star and an independent set, we determine the answer exactly.
In 2009, Bang-Jensen asked whether there exists a function $g(k)$ such that every strongly $k$-connected $n$-vertex tournament contains a strongly $k$-connected spanning subgraph with at most $kn + g(k)$ arcs. In this paper, we answer the question by showing that every strongly $k$-connected $n$-vertex tournament contains a strongly $k$-connected spanning subgraph with at most $kn + 750k^2\log(k+1)$ arcs.
We characterise the structure of those graphs of a given order which maximise the number of connected induced subgraphs for seven different graph classes, each with other prescribed parameters like minimum degree, independence number, vertex cover number, vertex connectivity, edge connectivity, chromatic number, number of bridges, thereby contributing to filling a gap in the literature.
Extremal graphs with minimum number of connected subgraphs in a given family
本文定义了顶点的子图数，刻画了在 $n$ 个顶点和 $k$ 个割点的所有图中具有最少连通子图数的极值图及其顶点，并在围长至少为 $k$ 的条件下得出相应结果，最终完成了对 $0 \leq k \leq 4$ 情形的完整刻画。
Characterizing forbidden induced subgraphs that force top vertices to be Gallai vertices
本文确定了使连通无导出-H图的每个顶顶点均为Gallai顶点的图H，结论为H是阶数至多4的线性森林或P₃+2P₁。
A strongly connected graph is strongly biconnected if after ignoring the direction of its edges we have an undirected graph with no articulation points. A 3-vertex strongly biconnected graph is a strongly biconnected digraph that has the property that deleting any two vertices in this graph leaves a strongly binconnected subgraph. Jaberi [11] presented approximation algorithms for minimum cardinality 2-vertex strongly biconnected directed subgraph problem. We will focus in this paper on polynomial time algorithms which we have implemented for producing spanning subgraphs that are 3-vertex stro…
A vertex whose removal in a graph $G$ increases the number of components of $G$ is called a cut vertex. For all $n,c$, we determine the maximum number of connected induced subgraphs in a connected graph with order $n$ and $c$ cut vertices, and also characterise those graphs attaining the bound. Moreover, we show that the cycle has the smallest number of connected induced subgraphs among all cut vertex-free connected graphs. The general case $c>0$ remains an open task. We also characterise the extremal graph structures given both order and number of pendant vertices, and establish the correspon…
Let $\hom(G)$ denote the size of the largest clique or independent set of a graph $G$. In 2007, Bukh and Sudakov proved that every $n$-vertex graph $G$ with $\hom(G) = O(\log n)$ contains an induced subgraph with $Ω(n^{1/2})$ distinct degrees, and raised the question of deciding whether an analogous result holds for every $n$-vertex graph $G$ with $\hom(G) = O(n^ε)$, where $ε> 0$ is a fixed constant. Here, we answer their question in the affirmative and show that every graph $G$ on $n$ vertices contains an induced subgraph with $Ω((n/\hom(G))^{1/2})$ distinct degrees. We also prove a stronger …
We prove that for all nonnegative integers k,s there exists c with the following property. Let G be a graph with clique number at most k and chromatic number more than c. Then for every vertex-colouring (not necessarily optimal) of G, some induced subgraph of G is an s-vertex path, and all its vertices have different colours. This extends a recent result of Gyarfas and Sarkozy, who proved the same (when k=2) for graphs G with girth at least five.
The Reconstruction Conjecture of Ulam asserts that, for $n\geq 3$, every $n$-vertex graph is determined by the multiset of its induced subgraphs with $n-1$ vertices. The conjecture is known to hold for various special classes of graphs but remains wide open. We survey results on the more general conjecture by Kelly from 1957 that for every positive integer $\ell$ there exists $M_\ell$ (with $M_1=3$) such that when $n\geq M_\ell$ every $n$-vertex graph is determined by the multiset of its induced subgraphs with $n-\ell$ vertices.
The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. Let $G$ be a connected graph constructed from pairwise disjoint connected graphs $G_1,\ldots ,G_k$ by selecting a vertex of $G_1$, a vertex of $G_2$, and identify these two vertices. Then continue in this manner inductively. We say that $G$ is obtained by point-attaching from $G_1, \ldots ,G_k$ and that $G_i$'s are the primary subgraphs of $G$. In this paper, we consider some particular …
An important theme of recent research in Ramsey theory has been establishing pseudorandomness properties of Ramsey graphs. An $N$-vertex graph is called $C$-Ramsey if it has no homogeneous set of size $C\log N$. A theorem of Bukh and Sudakov, solving a conjecture of Erdős, Faudree and Sós, shows that any $C$-Ramsey $N$-vertex graph contains an induced subgraph with $Ω_C(N^{1/2})$ distinct degrees. We improve this to $Ω_C(N^{2/3})$, which is tight up to the constant factor. We also show that any $N$-vertex graph with $N > (k-1)(n-1)$ and $n\geq n_0(k) = Ω(k^9)$ either contains a homogeneous set…
Large odd induced subgraphs via odd cuts
研究无孤立顶点的\(n\)顶点图，利用特定二分法，得出其有至少\(n/5\)个顶点且所有度数均为奇数的导出子图，改进了该类图的通用下界。
For a graph $G$, let $f_2(G)$ denote the largest number of vertices in a $2$-regular subgraph of $G$. We determine the minimum of $f_2(G)$ over $3$-regular $n$-vertex simple graphs $G$. To do this, we prove that every $3$-regular multigraph with exactly $c$ cut-edges has a $2$-regular subgraph that omits at most $\max\{0,\lfloor (c-1)/2\rfloor\}$ vertices. More generally, every $n$-vertex multigraph with maximum degree $3$ and $m$ edges has a $2$-regular subgraph that omits at most $\max\{0,\lfloor (3n-2m+c-1)/2\rfloor\}$ vertices. These bounds are sharp; we describe the extremal multigraphs.
Caterpillars with $n$ vertices are reconstructible from subgraphs with at most $n/2+1$ vertices
该研究证明了对于n≥48，任何n个顶点的毛虫可以由其m-图重建，且结果对Nýdl关于树的猜想提供了特例证明。
In 2007 Matamala proved that if $G$ is a simple graph with maximum degree $Δ\geq 3$ not containing $K_{Δ+1}$ as a subgraph and $s, t$ are positive integers such that $s+t \geq Δ$, then the vertex set of $G$ admits a partition $(S,T)$ such that $G[S]$ is a maximum order $(s-1)$-degenerate subgraph of $G$ and $G[T]$ is a $(t-1)$-degenerate subgraph of $G$. This result extended earlier results obtained by Borodin, by Bollobás and Manvel, by Catlin, by Gerencsér and by Catlin and Lai. In this paper we prove a hypergraph version of this result and extend it to variable degeneracy and to partitions …
Motivated by the concept of well-covered graphs, we define a graph to be well-bicovered if every vertex-maximal bipartite subgraph has the same order (which we call the bipartite number). We first give examples of them, compare them with well-covered graphs, and characterize those with small or large bipartite number. We then consider graph operations including the union, join, and lexicographic and cartesian products. Thereafter we consider simplicial vertices and 3-colored graphs where every vertex is in triangle, and conclude by characterizing the maximal outerplanar graphs that are well-bi…
On the typical structure of graphs not containing a fixed vertex-critical subgraph
本文研究稀疏无H图的典型结构，扩展相关开创性结果，解决Balogh等人关于严格2-平衡边临界图H的相变猜想，证明含所有边临界图的顶点临界图族中每个H的典型无H图结构均会经历相变。
Sparse vertex cutsets and the maximum degree
本文研究最大度受限大图中的稀疏顶点割集，证明存在小规模且诱导子图最大度显著低于 Δ 的割集，并在 Δ=4,5 及不含 K_{r,r} 子图时给出更强界。
A Roman dominating function (RD-function) on a graph $G = (V(G), E(G))$ is a labeling $f : V(G) \rightarrow \{0, 1, 2\}$ such that every vertex with label $0$ has a neighbor with label $2$. The weight $f(V(G))$ of a RD-function $f$ on $G$ is the value $Σ_{v\in V(G)} f (v)$. The {\em Roman domination number} $γ_{R}(G)$ of $G$ is the minimum weight of a RD-function on $G$. The six classes of graphs resulting from the changing or unchanging of the Roman domination number of a graph when a vertex is deleted, or an edge is deleted or added are considered. We consider relationships among the classes…
Let $α_{1} (G)$ denote the maximum size of an edge set that contains at most one edge from each triangle of $G$. Let $τ_{B} (G)$ denote the minimum size of an edge set whose deletion makes $G$ bipartite. It was conjectured by Lehel and independently by Puleo that $α_{1} (G) + τ_{B} (G) \le n^2/4$ for every $n$-vertex graph $G$. Puleo showed that $α_{1} (G) + τ_{B} (G) \le 5n^2/16$ for every $n$-vertex graph $G$. In this note, we improve the bound by showing that $α_{1} (G) + τ_{B} (G) \le 4403n^2/15000$ for every $n$-vertex graph $G$.
We define a $(V_1, V_2, \ldots, V_k)$-partition for a given graph $H$ and graphical properties $P_1, P_2, \ldots, P_k$ as a partition where each $V_i$ induces a subgraph of $H$ with property $P_i$. Matamala (2007) extended this result by showing that for any graph $H$ with $Δ(H)=p+q$, there exists a $(V_1, V_2)$-partition of $V(H)$ where $H[V_1]$ is a maximum order $(p-1)$-degenerate induced subgraph and $H[V_2]$ is $(q-1)$-degenerate. Additionally, Catlin and Lai proved that if $Δ(H)\geq 5$, $H$ has a $(V_1, V_2)$-partition such that $H[V_1]$ is a maximum order acyclic induced subgraph, $ω(H[…
We give a simple proof of a generalization of an inequality for homomorphism counts by Sidorenko (1994). A special case of our inequality says that if $d_v$ denotes the degree of a vertex $v$ in a graph $G$ and $\textrm{Hom}_Δ(H, G)$ denotes the number of homomorphisms from a connected graph $H$ on $h$ vertices to $G$ which map a particular vertex of $H$ to a vertex $v$ in $G$ with $d_v \ge Δ$, then $ \textrm{Hom}_Δ(H,G) \le \sum_{v\in G} d_v^{h-1}\mathbf{1}_{d_v\ge Δ} $ We use this inequality to study the minimum sample size needed to estimate the number of copies of $H$ in $G$ by sampling ve…
In this paper we completely resolve the well-known problem of Erdős and Sauer from 1975 which asks for the maximum number of edges an $n$-vertex graph can have without containing a $k$-regular subgraph, for some fixed integer $k\geq 3$. We prove that any $n$-vertex graph with average degree at least $C_k\log \log n$ contains a $k$-regular subgraph. This matches the lower bound of Pyber, Rödl and Szemerédi and substantially improves an old result of Pyber, who showed that average degree at least $C_k\log n$ is enough. Our method can also be used to settle asymptotically a problem raised by Erdő…
Reconfiguration of vertex colouring and forbidden induced subgraphs
本文研究受禁用导出子图限制的k-可着色图在(k+1)-着色重构图中的连通性与直径，刻画了单个禁用导出子图情形下连通性的充要条件，并给出了两类双禁用导出子图图类的连通性结果。
Maximizing the spectral radius of graphs of given size with forbidden a subgraph
本文研究边数固定且不含扇形图 $H_7$ 的图的最大谱半径，证明当 $m\geq33$ 时上界为 $1+\sqrt{m-2}$，并刻画唯一极值图。
A coloured graph is k-ultrahomogeneous if every isomorphism between two induced subgraphs of order at most k extends to an automorphism. A coloured graph is t-tuple regular if the number of vertices adjacent to every vertex in a set S of order at most k depends only on the isomorphism type of the subgraph induced by S. We classify the finite vertex-coloured k-ultrahomogeneous graphs and the finite vertex-coloured l-tuple regular graphs for k at least 4 and l at least 5, respectively. Our theorem in particular classifies finite vertex-coloured ultrahomogeneous graphs, where ultrahomogeneous mea…
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