(m,n)-混合图带切换的2-着色问题是多项式可解的

Hermitian adjacency spectrum and switching equivalence of mixed graphs

It is shown that an undirected graph $G$ is cospectral with the Hermitian adjacency matrix of a mixed graph $D$ obtained from a subgraph $H$ of $G$ by orienting some of its edges if and only if $H=G$ and $D$ is obtained from $G$ by a four-way switching operation; if $G$ is connected, this happens if and only if $λ_1(G)=λ_1(D)$. All mixed graphs of rank 2 are determined and this is used to classify which mixed graphs of rank 2 are cospectral with respect to their Hermitian adjacency matrix. Several families of mixed graphs are found that are determined by their Hermitian spectrum in the sense t…

Switching equivalence of Hermitian adjacency matrices of mixed graphs

Let $0 \in Γ$ and $Γ\setminus \{0\}$ be a subgroup of the complex numbers of unit modulus. Define $\mathcal{H}_{n}(Γ)$ to be the set of all $n\times n$ Hermitian matrices with entries in $Γ$, whose diagonal entries are zero. The matrices $A,B\in \mathcal{H}_{n}(Γ)$ are said to be switching equivalent if there is a diagonal matrix $D$, in which the diagonal entries belong to $Γ\setminus \{0\}$, such that $D^{-1} A D=B$. We find a characterization, in terms of fundamental cycles of graphs, of switching equivalence of matrices in $\mathcal{H}_{n}(Γ)$. We give sufficient conditions to characterize…

Switching $(m, n)$-mixed graphs with respect to Abelian groups

We extend results of Brewster and Graves for switching $m$-edge coloured graphs with respect to a cyclic group to switching $(m, n)$-mixed graphs with respect to an Abelian group. In particular, we establish the existence of a $(m, n)$-mixed graph $P_Γ(H)$ with the property that a $(m, n)$-mixed graph $G$ is switch equivalent to $H$ if and only if it is a special subgraph of $P_Γ(H)$, and the property that that $G$ can be switched to have a homomorphism to $H$ if and only if it has a homomorphism (without switching) to $P_Γ(H)$. We consider the question of deciding whether a $(m, n)$-mixed gra…

A rainbow blow-up lemma

We prove a rainbow version of the blow-up lemma of Komlós, Sárközy and Szemerédi for $μn$-bounded edge colourings. This enables the systematic study of rainbow embeddings of bounded degree spanning subgraphs. As one application, we show how our blow-up lemma can be used to transfer the bandwidth theorem of Böttcher, Schacht and Taraz to the rainbow setting. It can also be employed as a tool beyond the setting of $μn$-bounded edge colourings. Kim, Kühn, Kupavskii and Osthus exploit this to prove several rainbow decomposition results. Our proof methods include the strategy of an alternative proo…

$m$-边着色图上 $\Gamma$-可切换 $H$-着色的二分定理

A dichotomy theorem for $Γ$-switchable $H$-colouring on $m$-edge coloured graphs

研究 $m$-边着色图上通过切换操作实现到固定图 $H$ 的颜色保持同态的存在性，并在 $\Gamma$ 传递时给出二分定理。

The Ramsey number of mixed-parity cycles III

Denote by $R(G_1, G_2, G_3)$ the minimum integer $N$ such that any three-colouring of the edges of the complete graph on $N$ vertices contains a monochromatic copy of a graph $G_i$ coloured with colour $i$ for some $i\in{1,2,3}$. In a series of three papers of which this is the third, we consider the case where $G_1, G_2$ and $G_3$ are cycles of mixed parity. Specifically, in this in this paper, we consider $R(C_n,C_m,C_{\ell})$, where $n$ is even and $m$ and $\ell$ are odd. Figaj and Łuczak determined an asymptotic result for this case, which we improve upon to give an exact result. We prove …

The Simple Chromatic Number of $(m,n)$-Mixed Graphs

An $(m,n)$-mixed graph generalizes the notions of oriented graphs and edge-coloured graphs to a graph object with $m$ arc types and $n$ edge types. A simple colouring of such a graph is a non-trivial homomorphism to a reflexive target. We find that simple chromatic number of complete $(m,n)$-mixed graphs can be found in polynomial time. For planar graphs and $k$-trees ($k \geq 3$) we find that allowing the target to be reflexive does not lower the chromatic number of the respective family of $(m,n)$-mixed graphs. This implies that the search for universal targets for such families may be restr…

Hermitian adjacency matrices of mixed multigraphs

A mixed multigraph is obtained from an undirected multigraph by orienting a subset of its edges. In this paper, we study a new Hermitian matrix representation of mixed multigraphs, give an introduction to cospectral operations on mixed multigraphs, and characterize switching equivalent mixed multigraphs in terms of fundamental cycle basis. As an application, an upper bound of cospectral classes of mixed multigraphs with the same underlying graph is obtained.

The Ramsey number of mixed-parity cycles II

Denote by $R(G_1, G_2, G_3)$ the minimum integer $N$ such that any three-colouring of the edges of the complete graph on $N$ vertices contains a monochromatic copy of a graph $G_i$ coloured with colour $i$ for some $i\in{1,2,3}$. In a series of three papers of which this is the second, we consider the case where $G_1, G_2$ and $G_3$ are cycles of mixed parity. Here and in the previous paper, we consider $R(C_n,C_m,C_{\ell})$, where $n$ and $m$ are even and $\ell$ is odd. Figaj and Łuczak determined an asymptotic result for this case, which we improve upon to give an exact result. We prove that…

The Ramsey number of mixed-parity cycles I

Denote by $R(G_1, G_2, G_3)$ the minimum integer $N$ such that any three-colouring of the edges of the complete graph on $N$ vertices contains a monochromatic copy of a graph $G_i$ coloured with colour $i$ for some $i\in{1,2,3}$. In a series of three papers of which this is the first, we consider the case where $G_1, G_2$ and $G_3$ are cycles of mixed parity. Specifically, in this and the subsequent paper, we consider $R(C_n,C_m,C_{\ell})$, where $n$ and $m$ are even and $\ell$ is odd. Figaj and Łuczak determined an asymptotic result for this case, which we improve upon to give an exact result…

Hermitian adjacency matrix of the second kind for mixed graphs

This contribution gives an extensive study on spectra of mixed graphs via its Hermitian adjacency matrix of the second kind { ($N$-matrix for short)} introduced by Mohar \cite{0001}. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from $u$ to $v$ is equal to the sixth root of unity $ω=\frac{1+{\bf i}\sqrt{3}}{2}$ (and its symmetric entry is $\barω=\frac{1-{\bf i}\sqrt{3}}{2}$); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. The main results of this paper include the following: {equivalent} conditions for a mixed gr…

Mixed graphs with smallest eigenvalue greater than $-\frac{\sqrt{5}+1}{2}$

The classical problem of characterizing the graphs with bounded eigenvalues may date back to the work of Smith in 1970. Especially, the research on graphs with smallest eigenvalues not less than $-2$ has attracted widespread attention. Mixed graphs are natural generalization of undirected graphs. In this paper, we completely characterize the mixed graphs with smallest Hermitian eigenvalue greater than $-\frac{\sqrt{5}+1}{2}$, which consists of three infinite classes of mixed graphs and $30$ scattered mixed graphs. By the way, we get a new class of mixed graphs switching equivalent to their und…

$3$-Regular mixed graphs with optimum Hermitian energy

Let $G$ be a simple undirected graph, and $G^ϕ$ be a mixed graph of $G$ with the generalized orientation $ϕ$ and Hermitian-adjacency matrix $H(G^ϕ)$. Then $G$ is called the underlying graph of $G^ϕ$. The Hermitian energy of the mixed graph $G^ϕ$, denoted by $\mathcal{E}_H(G^ϕ)$, is defined as the sum of all the singular values of $H(G^ϕ)$. A $k$-regular mixed graph on $n$ vertices having Hermitian energy $n\sqrt{k}$ is called a $k$-regular optimum Hermitian energy mixed graph. In this paper, we first focus on the problem proposed by Liu and Li [J. Liu, X. Li, Hermitian-adjacency matrices and H…