Adam Kabela, Jakub Teska
We show that the square of every connected $S(K_{1,4})$-free graph satisfying a matching condition has a $2$-connected spanning subgraph of maximum degree at most~$3$. Furthermore, we characterise trees whose square has a $2$-connected spanning subgraph of maximum degree at most~$k$. This generalises the results on $S(K_{1,3})$-free graphs of Henry and Vogler (1985) and Harary and Schwenk (1971), respectively.
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In this work we investigate the chordality of squares and line graph squares of graphs. We prove a sufficient condition for the chordality of squares of graphs not containing induced cycles of length at least five. Moreover, we characterize the chordality of graph squares by forbidden subgraphs. Transferring that result to line graphs allows us to characterize the chordality of line graph squares.
We resolve a conjecture of Hegarty regarding the number of edges in the square of a regular graph. If $G$ is a connected $d$-regular graph with $n$ vertices, the graph square of $G$ is not complete, and $G$ is not a member of two narrow families of graphs, then the square of $G$ has at least $(2-o_d(1))n$ more edges than $G$.
Characterizing Graphs as Algebraic Squares
本文从图论视角刻画代数平方图，给出必要/充分条件并分类经典图族，证明联结与多种乘积保持平方性。
The square of a connected graph $G$ is obtained from $G$ by adding an edge between every pair of vertices at distance $2$. In this paper we give some upper or lower bounds for the spectral radius of the square of connected graphs, trees and unicyclic graphs respectively.We also investigate the spectral radius of the square of unicyclic graphs with given girth or trees with fixed diameter.
We introduce a graph attached to mutually orthogonal Sudoku Latin squares. The spectra of the graphs obtained from finite fields are explicitly determined. As a corollary, we then use the eigenvalues to distinguish non-isomorphic Sudoku Latin squares.
The chromatic number of a Latin square is the least number of partial transversals which cover its cells. This is just the chromatic number of its associated Latin square graph. Although Latin square graphs have been widely studied as strongly regular graphs, their chromatic numbers appear to be unexplored. We determine the chromatic number of a circulant Latin square, and find bounds for some other classes of Latin squares. With a computer, we find the chromatic number for all main classes of Latin squares of order at most eight.
We conjecture that the sum of the square roots of the Laplacian eigenvalues of a graph does not exceed the sum of the square roots of its signless Laplacian eigenvalues.
We prove the conjecture made by G.Wegner in 1977 that the square of every planar, cubic graph is $7$-colorable. Here, $7$ cannot be replaced by $6$.
In this paper we determine all the bipartite graphs with the maximum sum of squares of degrees among the ones with a given number of vertices and edges.
We show that under certain conditions the square of the graph obtained by identifying a vertex in two graphs with hamiltonian square is also hamiltonian. Using this result, we prove necessary and sufficient conditions for hamiltonicity of the square of a connected graph such that every vertex of degree at least three in a block graph corresponds to a cut vertex and any two these vertices are at distance at least four.
We show that the choosability of the square of planar graphs of max degree 4 without five cycles is at most 12. Keywords: planar graph, choosability AMS Mathematics Subject Classification: 05C15
We study equitable partitions of Latin-square graphs, and give a complete classification of those whose quotient matrix does not have an eigenvalue $-3$.
A graph is {\it square-complementary} ({\it squco}, for short) if its square and complement are isomorphic. We prove that there is no squco graph of girth $6$, thus answersing a question asked by Milani\vc et al. [Discrete Math., 2014, to appear], and leaving $g = 5$ as the only possible value of $g$ for which the existence of a squco graph of girth $g$ is unknown.
In combinatorics, a latin square is a $n\times n$ matrix filled with n different symbols, each occurring exactly once in each row and exactly once in each column. Associated to each latin square, we can define a simple graph called a latin square graph. In this article, we compute lower and upper bounds for the domination number and the k-tuple total domination numbers of such graphs. Moreover, we describe a formula for the 2-tuple total domination number.
In this paper we prove that balanced characteristic functions of canonical cliques in a Paley graph of square order $P(q^2)$ span the $\frac{-1+q}{2}$-eigenspace of the graph. This is the first of two steps to a second proof of the analogue of Erdös-Ko-Rado theorem for Paley graphs of square order (the first proof was given by A. Blokhuis in 1984).
In this paper we prove that the square of an essentially 2-edge connected graph with an additional property has a connected even factor with maximum degree at most 4. Moreover we show that, in general, the square of essentially 2-edge connected graph does not contain a connected even factor with bounded maximum degree.
Wegner conjectured in 1977 that the square of every planar graph with maximum degree at most $3$ is $7$-colorable. We prove this conjecture using the discharging method and computational techniques to verify reducible configurations.
We prove that if two graphs of girth at least 6 have isomorphic squares, then the graphs themselves are isomorphic. This is the best possible extension of the results of Ross and Harary on trees and the results of Farzad et al. on graphs of girth at least 7. We also make a remark on reconstruction of graphs from their higher powers.
Square coloring is a variant of graph coloring where vertices within distance two must receive different colors. When considering planar graphs, the most famous conjecture (Wegner, 1977) states that $\frac32Δ+1$ colors are sufficient to square color every planar graph of maximum degree $Δ$. This conjecture has been proven asymptotically for graphs with large maximum degree. We consider here planar graphs with small maximum degree and show that $2Δ+7$ colors are sufficient, which improves the best known bounds when $6\leqslant Δ\leqslant 31$.
On $α$-Square-Stable Graphs
图G的稳定数α（G）是图G中最大独立集的基数，图G中最大匹配的基数是图G中最大匹配的基数。如果alpha（G）+ mu（G）等于它的阶，则G是一个Koenig-Egervary图.称G为$α$ -平方稳定图，简称平方稳定图，如果alpha（G）= alpha（G*G），其中G*G表示G的二次幂.这些图首先由Randerath和Wolkmann研究。本文给出了平方稳定图的几个新的刻画。我们还证明了G是平方稳定的Koenig-Egervary图当且仅当它有一个由悬垂边组成的完美匹配。此外，我们发现，良好覆盖的树正是平方稳定的树。为了验证这一结果，我们给出了一个新的证明Ravindra的定理描述覆盖良好的树。
Characterizing and recognizing exact-distance squares of graphs
本文研究了图的精确距离平方的刻画与识别问题，给出了具有精确距离平方根的图的刻画及多项式时间识别算法，证明了二部精确距离平方根识别是 NP 完全的，并刻画了以树为精确距离平方根的图。
We generalize the notion of orthogonal latin squares to colorings of simple graphs. Two $n$-colorings of a graph are said to be \emph{orthogonal} if whenever two vertices share a color in one coloring they have distinct colors in the other coloring. We show that the usual bounds on the maximum size of a certain set of orthogonal latin structures such as latin squares, row latin squares, equi-$n$ squares, single diagonal latin squares, double diagonal latin squares, or sudoku squares are a special cases of bounds on orthogonal colorings of graphs.
Let $R_{n}$ denote the graph with vertex set consisting of the squares of an $n \times n$ grid, with two squares of the grid adjacent when they lie in the same row or column. This is the square rook's graph, and can also be thought of as the Cartesian product of two complete graphs of order $n$, or the line graph of the complete bipartite graph $K_{n,n}$. In this paper we compute the Smith group and critical group of the graph $R_{n}$ and its complement. This is equivalent to determining the Smith normal form of both the adjacency and Laplacian matrix of each of these graphs. In doing so we ve…
We explicitly describe the possible pairs of triangle and square densities for r-regular finite simple graphs. We also prove that every r-regular unimodular random graph can be approximated by r-regular finite graphs with respect to these densities. As a corollary one gets an explicit description of the possible pairs of the third and fourth moments of the spectral measure of r-regular unimodular random graphs.
We prove a conjecture of Dvořák, Král, Nejedlý, and Škrekovski that planar graphs of girth at least five are square $(Δ+2)$-colorable for large enough $Δ$. In fact, we prove the stronger statement that such graphs are square $(Δ+2)$-choosable and even square $(Δ+2)$-paintable.
We consider the class of Berge graphs that contain no odd prism and no square (cycle on four vertices). We prove that every graph G in this class either is a clique or has an even pair, as conjectured by Everett and Reed. This result is used to devise a polynomial-time algorithm to color optimally every graph in this class.
We show that the threshold for the random graph $G_{n,p}$ to contain the square of a Hamilton cycle is $p=\frac{1}{\sqrt{n}}$. This improves the previous results of Kühn and Osthus and also Nenadov and Škorić. In addition we consider how many random edges need to be added to a graph of order $n$ with minimum degree $αn$ in order that it contains the square of a Hamilton cycle w.h.p.
Square-free Word-representation of Word-representable Graphs
本文证明了任何非空词可表示图均可用无平方词表示，并探讨了无平方词表示在图论中的应用与计数。
We study a metric on the set of finite graphs in which two graphs are considered to be similar if they have similar bounded dimensional "factors". We show that limits of convergent graph sequences in this metric can be represented by symmetric Borel measures on $[0,1]^2$. This leads to a generalization of dense graph limit theory to sparse graph sequences.
The square of a graph is obtained by adding additional edges joining all pair of vertices of distance two in the original graph. Particularly, if $C$ is a hamiltonian cycle of a graph $G$, then the square of $C$ is called a hamiltonian square of $G$. In this paper, we characterize all possible forbidden pairs, which implies the containment of a hamiltonian square, in a 4-connected graph. The connectivity condition is necessary as, except $K_3$ and $K_4$, the square of a cycle is always 4-connected.
The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. Let $χ(H)$ and $χ_{\ell}(H)$ be the chromatic number and the list chromatic number of $H$, respectively. A graph $H$ is called {\em chromatic-choosable} if $χ_{\ell} (H) = χ(H)$. It is an interesting problem to find graphs that are chromatic-choosable. Motivated by the List Total Coloring Conjecture, Kostochka and Woodall (2001) proposed the List Square Coloring Conjecture which states that $G^2$ is chromatic-choosable …
We give bounds on the L(2,1)-labeling number of a simple graph in terms of its order and its maximum degree. We also describe an infinite class of graphs of which the elements have the highest L(2,1)-labeling numbers in terms of their maximum degrees of any known infinite class of graphs.
Coloring Square-free Berge Graphs
针对不含长度为4的无弦环的Berge图类，提出一种纯图论算法，可在多项式时间内实现最优着色。
In 1977, Wegner conjectured that the chromatic number of the square of every planar graph $G$ with maximum degree $Δ\ge8$ is at most $\bigl\lfloor\frac32Δ\bigr\rfloor+1$. We show that it is at most $\frac32 Δ(1+o(1))$ (where the $o(1)$ is as $Δ\to+\infty$), and indeed that this is true for the list chromatic number and for more general classes of graphs.
In this paper we introduce new models of random graphs, arising from Latin squares which include random Cayley graphs as a special case. We investigate some properties of these graphs including their clique, independence and chromatic numbers, their expansion properties as well as their connectivity and Hamiltonicity. The results obtained are compared with other models of random graphs and several similarities and differences are pointed out. For many properties our results for the general case are as strong as the known results for random Cayley graphs and sometimes improve the previously bes…
Nowhere-zero $3$-flows in Cayley graphs on solvable groups of twice square-free order
本文验证了Tutte的3-流猜想，针对Cayley图在可解群二次平方自由阶上的情况，提出新的必要充分条件以确定简单5度图是否具有无零3-流。
We consider latin square graphs $Γ= \rm{LSG}(H)$ of the Cayley table of a given finite group $H$. We characterize all pairs $(Γ,G)$, where $G$ is a subgroup of autoparatopisms of the Cayley table of $H$ such that $G$ acts arc-transitively on $Γ$ and all nontrivial $G$-normal quotient graphs of $Γ$ are complete. We show that $H$ must be elementary abelian and determine the number $k$ of complete normal quotients. This yields new infinite families of diameter two arc-transitive graphs with $k = 1$ or $k = 2$.
We show that $p=\sqrt{\frac{e}{n}}$ is a sharp threshold for the random graph $G_{n,p}$ to contain the square of a Hamilton cycle. This improves the previous results of Kühn and Osthus and also Nenadov and Škorić.
On Colorings of Squares of Outerplanar Graphs
研究了外平面图$G$的正方形$G^2$的顶点着色.对于所有的$Δ\in \nats$，我们找到了$G^2$的诱导性、色数和团数的最优界，它是$G$的最大度$Δ$的函数.作为奖励，我们得到了当$G^2$.在弦外平面图的情况下，我们准确地分类哪些图的参数超过了绝对极小值。
Establishing inequalities among graph densities is a central pursuit in extremal combinatorics. A standard tool to certify the nonnegativity of a graph density expression is to write it as a sum of squares. In this paper, we identify a simple condition under which a graph density expression cannot be a sum of squares. Using this result, we prove that the Blakley-Roy inequality does not have a sum of squares certificate when the path length is odd. We also show that the same Blakley-Roy inequalities cannot be certified by sums of squares using a multiplier of the form one plus a sum of squares.…
The most general structure of graphs with hamiltonian or hamiltonian connected square
本文刻画使图平方 $G^2$ 为哈密顿图或哈密顿连通图的最一般块-割点结构，并借鉴哈密顿全图的研究思路。
An \emph{edge coloring} of a graph $G$ is strong if each color class is an induced matching of $G$. The \emph{strong chromatic index} of $G$, denoted by $χ_{s}^{\prime }(G)$, is the minimum number of colors for which $G$ has a strong edge coloring. The strong chromatic index of $G$ is equal to the chromatic number of the square of the line graph of $G$. The chromatic number of the square of the line graph of $G$ is greater than or equal to the clique number of the square of the line graph of $G$, denoted by $ω(L)$. In this note we prove that $ω(L) \le 1.5 Δ_{G}^2$ for every graph $G$. Our resu…
We study squares of planar graphs with the aim to determine their list chromatic number. We present new upper bounds for the square of a planar graph with maximum degree $Δ\leq 4$. In particular $G^2$ is 5-, 6-, 7-, 8-, 12-, 14-choosable if the girth of $G$ is at least 16, 11, 9, 7, 5, 3 respectively. In fact we prove more general results, in terms of maximum average degree, that imply the results above.
A well known upper bound for the spectral radius of a graph, due to Hong, is that $μ_1^2 \le 2m - n + 1$. It is conjectured that for connected graphs $n - 1 \le s^+ \le 2m - n + 1$, where $s^+$ denotes the sum of the squares of the positive eigenvalues. The conjecture is proved for various classes of graphs, including bipartite, regular, complete $q$-partite, hyper-energetic, and barbell graphs. Various searches have found no counter-examples. The paper concludes with a brief discussion of the apparent difficulties of proving the conjecture in general.
A graph K is square-free if it contains no four-cycle as a subgraph. A graph K is multiplicative if GxH -> K implies G -> K or H -> K, for all graphs G,H. Here GxH is the tensor (or categorical) graph product and G -> K denotes the existence of a graph homomorphism from G to K. Hedetniemi's conjecture states that all cliques K_n are multiplicative. However, the only non-trivial graphs known to be multiplicative are K_3, odd cycles, and still more generally, circular cliques $K_{p/q}$ with 2 <= p/q < 4. We make no progress for cliques, but show that all square-free graphs are multiplicative. In…
Let $t(H;G)$ be the homomorphism density of a graph $H$ into a graph $G$. Sidorenko's conjecture states that for any bipartite graph $H$, $t(H;G)\geq t(K_2;G)^{|E(H)|}$ for all graphs $G$. It is already known that such inequalities cannot be certified through the sums of squares method when $H$ is a so-called trivial square. In this paper, we investigate recent results about Sidorenko's conjecture and classify those involving trivial versus non-trivial squares. We then present some computational results. In particular, we categorize the bipartite graphs $H$ on at most 7 edges for which $t(H;G)…
Amorphic association schemes with negative Latin square type graphs
应用偏差集、二次型的结果以及Brouwer和Van Dam的最新结果，构造了第一个已知的负拉丁方型图的无定形结合方案，其基础集是一个非初等交换2-群.给出了Hamilton的一个结果的简单证明，推广了Brouwer的结果.我们使用多个不同的二次型来构造具有大量类的无定形结合方案。
The Thue number $π(G)$ of a graph $G$ is the minimum number of colors needed to color $G$ without creating a square on a path of $G$. For a graph class $C$, $π(C)$ is the supremum of $π(G)$ over the graphs $G\in C$. The Thue number has been investigated for famous minor-closed classes: $π(tree)=4$, $7\leπ(outerplanar)\le12$, and $11\leπ(planar)\le768$. Following a suggestion of Grytczuk, we consider the generalized parameters $π_k(C)$ such that only squares of period at least $k$ must be avoided. Thus, $π(C)=π_1(C)$. We show that $π_5(tree)=2$, $π_2(tree)=3$, and $π_k(planar)\ge11$ for every f…
Positive Square Energy of Graphs with Minimum Degree at Least Two
本文针对最小度至少为2的连通图，将平方能量下界从$n-1$加强到$n$，除非该图是圈。
Symmetry and asymmetry between positive and negative square energies of graphs
本文研究了图的正负平方能量 $s^+(G)$ 和 $s^-(G)$ 之间的对称性与非对称性，通过大型随机图等新例证回答了既有问题，并提出了未来研究方向。
Coloring, List Coloring, and Painting Squares of Graphs (and other related problems)
本文综述了图的平方的着色、列表着色和涂色问题，特别是强边着色，主要关注平面图和其他稀疏图类。
We prove that a family of pseudo-Paley graphs of square order obtained from unions of cyclotomic classes satisfies the Erdős-Ko-Rado (EKR) module property, in a sense that the characteristic vector of each maximum clique is a linear combination of characteristic vectors of canonical cliques. This extends the EKR-module property of Paley graphs of square order and solves a problem proposed by Godsil and Meagher. Different from previous works, which heavily rely on tools from number theory, our approach is purely combinatorial in nature. The main strategy is to view these graphs as block graphs …
In the past various distance based colorings on planar graphs were introduced. We turn our focus to three of them, namely $2$-distance coloring, injective coloring, and exact square coloring. A $2$-distance coloring is a proper coloring of the vertices in which no two vertices at distance $2$ receive the same color, an injective coloring is a coloring of the vertices in which no two vertices with a common neighbor receive the same color, and an exact square coloring is a coloring of the vertices in which no two vertices at distance exactly $2$ receive the same color. We prove that planar graph…
In 1962, Pósa conjectured that a graph $G=(V, E)$ contains a square of a Hamiltonian cycle if $δ(G)\ge 2n/3$. Only more than thirty years later Komlós, Sárkőzy, and Szemerédi proved this conjecture using the so-called Blow-Up Lemma. Here we extend their result to a random graph setting. We show that for every $ε> 0$ and $p=n^{-1/2+ε}$ a.a.s. every subgraph of $G_{n,p}$ with minimum degree at least $(2/3+ε)np$ contains the square of a cycle on $(1-o(1))n$ vertices. This is almost best possible in three ways: (1) for $p\ll n^{-1/2}$ the random graph will not contain any square of a long cycle (2…
We provide new evidence that spanning forests of graphs satisfy the same negative correlation properties as spanning trees, derived from Lord Rayleigh's monotonicity property for electrical networks. The main result of this paper is that the Rayleigh difference for the spanning forest generating polynomial of a series parallel graph can be expressed as a certain positive sum of monomials times squares of polynomials. We also show that every regular matroid is independent-set-Rayleigh if and only if every basis-Rayleigh binary matroid is also independent-set-Rayleigh.
Interval edge-colorings of Cartesian products of graphs II
本文研究了图笛卡尔积的区间边着色，给出了W(G□H)的精确下界及连通图的新上界，并改进了多维环面和Hamming图的已知界限，同时探讨了Fibonacci立方体的区间着色。
A graph is said to be symmetric if its automorphism group is transitive on its arcs. Guo et al. (Electronic J. Combin. 18, \#P233, 2011) and Pan et al. (Electronic J. Combin. 20, \#P36, 2013) determined all pentavalent symmetric graphs of order $4pq$. In this paper, we shall generalize this result by determining all connected pentavalent symmetric graphs of order four times an odd square-free integer. It is shown in this paper that, for each of such graphs $\itΓ$, either the full automorphism group ${\sf Aut}\itΓ$ is isomorphic to ${\sf PSL}(2,p)$, ${\sf PGL}(2,p)$, ${\sf PSL}(2,p){\times}\mat…
In this paper we show that for integers $s\geq2$, $t\geq1$, any co-edge-regular graph which is cospectral with the $s$-clique extension of the $t\times t$-grid is the $s$-clique extension of the $t\times t$-grid, if $t$ is large enough. Gavrilyuk and Koolen used a weaker version of this result to show that the Grassmann graph $J_q(2D,D)$ is characterized by its intersection array as a distance-regular graph, if $D$ is large enough.
The Alon-Tarsi Number of Squares of Subcubic Planar Graphs without Cycles of Lengths $4$ to $8$
研究平面图平方的阿隆 - 塔尔西数，通过图多项式定义，Kim和Luo证明特定次立方平面图平方的列表色数≤6，本文强化结果，证明此类图平方的阿隆 - 塔尔西数≤6。
For any positive integer $s$, a $[2,2s]$-factor in a graph $G$ is a connected even factor with maximum degree at most $2s$. We prove that if every induced $S(K_{1, 2s+1})$ in a graph $G$ has at least 3 edges in a block of degree at most two, then $G^2$ has a $[2,2s]$-factor. This extends the results of Hendry and Vogler and of Abderrezzak et al.
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