A note on uniquely 10-colorable graphs

On uniquely k-list colorable planar graphs, graphs on surfaces, and regular graphs

A graph $G$ is called uniquely k-list colorable (U$k$LC) if there exists a list of colors on its vertices, say $L=\lbrace S_v \mid v \in V(G) \rbrace $, each of size $k$, such that there is a unique proper list coloring of $G$ from this list of colors. A graph $G$ is said to have property $M(k)$ if it is not uniquely $k$-list colorable. Mahmoodian and Mahdian characterized all graphs with property $M(2)$. For $k\geq 3$ property $M(k)$ has been studied only for multipartite graphs. Here we find bounds on $M(k)$ for graphs embedded on surfaces, and obtain new results on planar graphs. We begin a…

唯一可区分着色图

Uniquely Distinguishing Colorable Graphs

本文定义并研究唯一可区分着色图，提出1型与2型分类，刻画非连通图、树及二部图结构，并据此确定 G∪G 与 G 的可区分色数关系。

Some concepts in list coloring

In this paper uniquely list colorable graphs are studied. A graph G is called to be uniquely k-list colorable if it admits a k-list assignment from which G has a unique list coloring. The minimum k for which G is not uniquely k-list colorable is called the M-number of G. We show that every triangle-free uniquely vertex colorable graph with chromatic number k+1, is uniquely k-list colorable. A bound for the M-number of graphs is given, and using this bound it is shown that every planar graph has M-number at most 4. Also we introduce list criticality in graphs and characterize all 3-list critica…

Algebraic Characterization of Uniquely Vertex Colorable Graphs

The study of graph vertex colorability from an algebraic perspective has introduced novel techniques and algorithms into the field. For instance, it is known that $k$-colorability of a graph $G$ is equivalent to the condition $1 \in I_{G,k}$ for a certain ideal $I_{G,k} \subseteq \k[x_1, ..., x_n]$. In this paper, we extend this result by proving a general decomposition theorem for $I_{G,k}$. This theorem allows us to give an algebraic characterization of uniquely $k$-colorable graphs. Our results also give algorithms for testing unique colorability. As an application, we verify a counterexamp…

唯一可着色超图

Uniquely colorable hypergraphs

本文确定了所有 k≥r≥3 时唯一可着色超图阈值 Φ_{k,r} 的精确值，并发现其在约 k=(4r-2)/3 处出现图情形中不存在的相变现象，同时结合经典定理给出了最小正 i-度问题的一般紧界。

A note on 3-colorable plane graphs without 5- and 7-cycles

Borodin et al figured out a gap of the paper published at J. Combinatorial Theory Ser. B (Vol.96 (2006) 958--963), and gave a new proof with the similar technique. The purpose of this note is to fix the gap by slightly revising the definition of special faces, and adding a few lines of explanation in the proofs (new added text are all in black font).

A note on shortest circuit cover of 3-edge colorable cubic signed graphs

A {sign-circuit cover} $\mathcal{F}$ of a signed graph $(G, σ)$ is a family of sign-circuits which covers all edges of $(G, σ)$. The shortest sign-circuit cover problem was initiated by Má$\check{\text{c}}$ajová, Raspaud, Rollová, and Škoviera (JGT 2016) and received many attentions in recent years. In this paper, we show that every flow-admissible 3-edge colorable cubic signed graph $(G, σ)$ has a sign-circuit cover with length at most $\frac{20}{9} |E(G)|$.

A note on the locally irregular edge colorings of cacti

A graph is locally irregular if the degrees of the end-vertices of every edge are distinct. An edge coloring of a graph G is locally irregular if every color induces a locally irregular subgraph of G. A colorable graph G is any graph which admits a locally irregular edge coloring. The locally irregular chromatic index X'irr(G) of a colorable graph G is the smallest number of colors required by a locally irregular edge coloring of G. The Local Irregularity Conjecture claims that all colorable graphs require at most 3 colors for a locally irregular edge coloring. Recently, it has been observed t…

A note on equitable colorings of forests

This note gives a short proof on characterizations of a forest to be equitably k-colorable.

Every toroidal graphs without adjacent triangles is odd 8-colorable

Odd coloring is a proper coloring with an additional restriction that every non-isolated vertex has some color that appears an odd number of times in its neighborhood. The minimum number of colors $k$ that can ensure an odd coloring of a graph $G$ is denoted by $χ_o(G)$. We say $G$ is odd $k$-colorable if $χ_o(G)\le k$. This notion is introduced very recently by Petruševski and Škrekovski, who proved that if $G$ is planar then $ χ_{o}(G) \leq 9 $. A toroidal graph is a graph that can be embedded on a torus. Note that a $K_7$ is a toroidal graph, $χ_{o}(G)\leq7$. Tian and Yin proved that every …

Every toroidal graph without $3$-cycles is odd $7$-colorable

Odd coloring is a proper coloring with an additional restriction that every non-isolated vertex has some color that appears an odd number of times in its neighborhood. The minimum number of colors $k$ that can ensure an odd coloring of a graph $G$ is denoted by $χ_o(G)$. We say $G$ is $k$-colorable if $χ_o(G)\le k$. This notion is introduced very recently by Petruševski and Škrekovski, who proved that if $G$ is planar then $ χ_{o}(G) \leq 9 $. A toroidal graph is a graph that can be embedded on a torus. Note that a $K_7$ is a toroidal graph, $χ_{o}(G)\leq7$. In this paper, we proved that, ever…

强集合可着色图：一个完整的刻画

Strongly Set-Colorable Graphs: A Complete Characterization

本文给出强集合可着色图的完整刻画，证明其等价于关联 3-一致超图构成唯一 Steiner 三元系的 packing，从而统一并简化了该图类的结构理论。

关于移位链多色着色的一则注记

A Note on Polychromatic Colorings of Shift-Chains

该注记推广了大m下m-均匀移位链超图的正常2着色猜想，并证明对任意m都存在无法多色3着色的m-均匀移位链。

Note on 3-Choosability of Planar Graphs with Maximum Degree 4

Deciding whether a planar graph (even of maximum degree $4$) is $3$-colorable is NP-complete. Determining subclasses of planar graphs being $3$-colorable has a long history, but since Grötzsch's result that triangle-free planar graphs are such, most of the effort was focused to solving Havel's and Steinberg's conjectures. In this paper, we prove that every planar graph of maximum degree $4$ obtained as a subgraph of the medial graph of any bipartite plane graph is $3$-choosable. These graphs are allowed to have close triangles (even incident), and have no short cycles forbidden, hence represen…

A note on the incidence coloring of outerplanar graphs

A proof that every outerplanar graph is Δ+2 colorable. This is slightly stronger then an unpublished result of Wang Shudong, Ma Fangfang, Xu Jin, and Yan Lijun proving the same for 2-connected outerplanar graphs.

Note on $4$-coloring $6$-regular triangulations on the torus

In 1973, Altshuler characterized the $6$-regular triangulations on the torus to be precisely those that are obtained from a regular triangulation of the $r \times s$ toroidal grid where the vertices in the first and last column are connected by a shift of $t$ vertices. Such a graph is denoted $T(r, s, t)$. In 1999, Collins and Hutchinson classified the $4$-colorable graphs $T(r, s, t)$ with $r, s \geq 3$. In this paper, we point out a gap in their classification and show how it can be fixed. Combined with the classification of the $4$-colorable graphs $T(1, s, t)$ by Yeh and Zhu in 2003, this …

Three Colorability of an Arrangement Graph of Great Circles

Stan Wagon asked the following in 2000. Is every zonohedron face 3-colorable when viewed as a planar map? An equivalent question, under a different guise, is the following: is the arrangement graph of great circles on the sphere always vertex 3-colorable? (The arrangement graph has a vertex for each intersection point, and an edge for each arc directly connecting two intersection points.) Assume that no three circles meet at a point, so that this arrangement graph is 4-regular. In this note we have shown that all arrangement graphs defined as above are 3-colorable.

On Interval Non-Edge-Colorable Eulerian Multigraphs

An edge-coloring of a multigraph $G$ with colors $1,\ldots,t$ is called an interval $t$-coloring if all colors are used, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. In this note, we show that all Eulerian multigraphs with an odd number of edges have no interval coloring. We also give some methods for constructing of interval non-edge-colorable Eulerian multigraphs.

关于唯一可嵌入2-因子的一个注记

A note on uniquely embeddable 2-factors

本文研究由互不相交环组成的2-因子在其补图中的唯一嵌入性，并给出了这类2-因子的完整刻画。

On snarks that are far from being 3-edge colorable

In this note we construct two infinite snark families which have high oddness and low circumference compared to the number of vertices. Using this construction, we also give a counterexample to a suggested strengthening of Fulkerson's conjecture by showing that the Petersen graph is not the only cyclically 4-edge connected cubic graph which require at least five perfect matchings to cover its edges. Furthermore the counterexample presented has the interesting property that no 2-factor can be part of a cycle double cover.

An extension of Thomassen's result on choosability

Thomassen proved that all planar graphs are $5$-choosable. Škrekovski strengthened the result by showing that all $K_{5}$-minor-free graphs are $5$-choosable. Dvořák and Postle pointed out that all planar graphs are DP-$5$-colorable. In this note, we first improve these results by showing that every $K_{5}$-minor-free or $K_{3, 3}$-minor-free graph is DP-$5$-colorable. In the final section, we further improve these results under the term strictly $f$-degenerate transversal.

Existence of a small cover over a 15-colorable simple 4-polytope

The chromatic number for properly colouring the facets of a combinatorial simple $n$-polytope $P^n$ that is the orbit space of a quasitoric manifold satisfies the inequality $n\leq P^n\leq 2^n-1$. The inequality is sharp for $n=2$ but not for $n=3$ due to the Four Color theorem. In this note, we construct a simple 4-polytope admitting a characteristic map whose chromatic number equals $15$ and deduce that the predicted upper bound is attained for $n=4$. Analogues results are verified for the case of oriented small covers in dimensions $4$ and $5$.

关于两个拟阵的列表色数的一个注记

A note on the list chromatic number of two matroids

本文研究两个拟阵的公共独立集的列表着色，构造例子表明公共色数与公共列表色数可不等，并证明两个强基序拟阵若各自2-可着色则其交集2-列表可着色。

Uniquely $D$-colourable digraphs with large girth II: simplification via generalization

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 res…

A note on random greedy coloring of uniform hypergraphs

The smallest number of edges forming an n-uniform hypergraph which is not r-colorable is denoted by m(n,r). Erdős and Lovász conjectured that m(n,2)=θ(n 2^n)$. The best known lower bound m(n,2)=Ω(sqrt(n/log(n)) 2^n) was obtained by Radhakrishnan and Srinivasan in 2000. We present a simple proof of their result. The proof is based on analysis of random greedy coloring algorithm investigated by Pluhár in 2009. The proof method extends to the case of r-coloring, and we show that for any fixed r we have m(n,r)=Ω((n/log(n))^(1-1/r) r^n) improving the bound of Kostochka from 2004. We also derive ana…

Maximum average degree and relaxed coloring

We say a graph is $(d, d, \ldots, d, 0, \ldots, 0)$-colorable with $a$ of $d$'s and $b$ of $0$'s if $V(G)$ may be partitioned into $b$ independent sets $O_1,O_2,\ldots,O_b$ and $a$ sets $D_1, D_2,\ldots, D_a$ whose induced graphs have maximum degree at most $d$. The maximum average degree, $mad(G)$, of a graph $G$ is the maximum average degree over all subgraphs of $G$. In this note, for nonnegative integers $a, b$, we show that if $mad(G)< \frac{4}{3}a + b$, then $G$ is $(1_1, 1_2, \ldots, 1_a, 0_1, \ldots, 0_b)$-colorable.

The odd chromatic number of a toroidal graph is at most 9

It's well known that every planar graph is $4$-colorable. A toroidal graph is a graph that can be embedded on a torus. It's proved that every toroidal graph is $7$-colorable. A proper coloring of a graph is called \emph{odd} if every non-isolated vertex has at least one color that appears an odd number of times in its neighborhood. The smallest number of colors that admits an odd coloring of a graph $ G $ is denoted by $χ_{o}(G)$. In this paper, we prove that if $G$ is tortoidal, then $χ_{o}\left({G}\right)\le9$; Note that $K_7$ is a toroidal graph, the upper bound is no less than $7$.

关于4-正则图的$\bar{X}$-染色与$\hat{A}$-染色的注记

A note on $\bar{X}$-coloring and $\hat{A}$-coloring 4-regular graphs

本文研究了4-正则图的$\bar{X}$-染色与$\hat{A}$-染色问题，通过两种独立算法验证反例图，否定了Malnegro和Ozeki提出的两个问题及相关猜想。

Smaller counterexamples to Hedetniemi's conjecture

Hedetniemi's conjecture~\cite{hedetniemi1966homomorphisms} for $c$-colorings states that the tensor product $G \times H$ is $c$-colorable if and only if $G$ or $H$ is $c$-colorable. El-Zahar and Sauer~\cite{El-ZaharS85} proved it for $c = 3$. In a recent breakthrough, Shitov~\cite{Shitov19} showed counterexamples, for large $c$. While Shitov's proof is already remarkably short, Zhu \cite{Zhu20} simplified the argument and gave a more explicit counterexample for $c=125$. Tardif \cite{Tardif20} showed that a modification of the arguments allows to use ``wide colorings'' to obtain counterexamples…

On a certain representation of the chromatic polynomial

The representation is essentially the same as that given by J.P.Nagle in J. Comb. Theory (B), 1971, 10:1, 42--59. The distinction is in the definition of the weighting function via the number of flows. This new definition allows one to deduce a number of corollaries, in particular, the following. A) The chromatic polynomial of a connected planar graph G can be uniquely determined from its combinatory dual graph G^* (although the graph G itself isn't, in general, determined uniquely by G^*). B) If a planar graph G is different from the full graph K_3 and has exactly one (up to renaming of color…

The proof of Steinberg's three coloring conjecture

The well-known Steinberg's conjecture asserts that any planar graph without 4- and 5-cycles is 3 colorable. In this note we have given a short algorithmic proof of this conjecture based on the spiral chains of planar graphs proposed in the proof of the four color theorem by the author in 2004.

On-line list coloring of matroids

A coloring of a matroid is proper if elements of the same color form an independent set. A theorem of Seymour asserts that a k-colorable matroid is also colorable from any lists of size k. In this note we generalize this theorem to the on-line setting. We prove that a coloring of a matroid from lists of size k is possible even if appearances of colors in the lists are recovered color by color by an adversary, while our job is to assign a color immediately after it is recovered. We also prove a more general weighted version of our result with lists of varying sizes. In consequence we get a simp…

Universal completability, least eigenvalue frameworks, and vector colorings

An embedding $i \mapsto p_i\in \mathbb{R}^d$ of the vertices of a graph $G$ is called universally completable if the following holds: For any other embedding $i\mapsto q_i~\in \mathbb{R}^{k}$ satisfying $q_i^T q_j = p_i^T p_j$ for $i = j$ and $i$ adjacent to $j$, there exists an isometry mapping the $q_i$'s to the $p_i$'s for all $ i\in V(G)$. The notion of universal completability was introduced recently due to its relevance to the positive semidefinite matrix completion problem. In this work we focus on graph embeddings constructed using the eigenvectors of the least eigenvalue of the adjace…

Ore's Conjecture for $k=4$ and Gr\" otzsch Theorem

A graph $G$ is $k$-{\em critical} if it has chromatic number $k$, but every proper subgraph of $G$ is $(k-1)$--colorable. Let $f_k(n)$ denote the minimum number of edges in an $n$-vertex $k$-critical graph. In a very recent paper, we gave a lower bound, $f_k(n) \geq F(k,n)$, that is sharp for every $n=1\,({\rm mod}\, k-1)$. It is also sharp for $k=4$ and every $n\geq 6$. In this note, we present a simple proof of the bound for $k=4$. It implies the case $k=4$ of the conjecture by Ore from 1967 that for every $k\geq 4$ and $n\geq k+2$, $f_k(n+k-1)=f(n)+\frac{k-1}{2}(k - \frac{2}{k-1})$. We also…

关于遗传超图族多色染色的注记

Note on polychromatic coloring of hereditary hypergraph families

本文展示了一个5一致超图反驳了Keszegh和作者关于多色染色的猜想，推进了对遗传超图族多色染色的理解，并指出该方法不适用于任意高一致性超图。

Combinatorial Nullstellensatz and DP-coloring of Graphs

We initiate the study of applying the Combinatorial Nullstellensatz to the DP-coloring of graphs even though, as is well-known, the Alon-Tarsi theorem does not apply to DP-coloring. We define the notion of good covers of prime order which allows us to apply the Combinatorial Nullstellensatz to DP-coloring. We apply these tools to DP-coloring of the cones of certain bipartite graphs and uniquely 3-colorable graphs. We also extend a result of Akbari, Mirrokni, and Sadjad (2006) on unique list colorability to the context of DP-coloring. We establish a sufficient algebraic condition for a graph $G…

A note on the positive semidefinitness of $A_α(G)$

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For every real $α\in\left[ 0,1\right] $, write $A_α\left( G\right) $ for the matrix \[ A_α\left( G\right) =αD\left( G\right) +(1-α)A\left( G\right) . \] Let $α_{0}\left( G\right) $ be the smallest $α$ for which $A_α(G)$ is positive semidefinite. It is known that $α_{0}\left( G\right) \leq1/2$. The main results of this paper are: (1) if $G$ is $d$-regular then \[ α_{0}=\frac{-λ_{\min}(A(G))}{d-λ_{\min}(A(G))}, \] where $λ_{\min}(A(G))$ is the smallest eigenvalue of $A(G)$; (2) $G$ contai…

关于距离-衰减着色的一个注记

A Note on Distance-Fall Colorings

本文研究图的距离-衰减着色问题，证明了连通可3-着色图具有距离-2衰减3-着色，并由此证明了Beineke和Henning关于树中独立距离-d-支配集大小上界的猜想。

论有补格、唯一有补格与唯一有补非分配格（关于一个数学之谜的历史与认识论注记）

On complemented, uniquely complemented and uniquely complemented nondistributive lattices (a historical and epistemological note about a mathematical mystery)

本文从历史与认识论角度梳理有补格和唯一有补格的研究脉络，重点考察唯一有补非分配格这一数学之谜，并展示如何利用自由格技术构造此类结构。

关于$k$-着色图的一个注记

A note on graphs of $k$-colourings

本文证明每个图都可由某个$k$-着色图唯一确定，证实相关猜想，并表明有限个广义色多项式无法唯一反推原图。

Interval edge-colorings of K_{1,m,n}

In this note we prove that K_{1,m,n} is interval edge-colorable if and only if gcd(m+1,n+1)=1. It settles in the affirmative a conjecture of Petrosyan.

Weakening Total Coloring Conjecture: Weak TCC and Hadwiger's Conjecture on Total Graphs

Hadwiger's conjecture is one of the most important and long-standing conjectures in graph theory. Reed and Seymour showed in 2004 that Hadwiger's conjecture is true for line graphs. We investigate this conjecture on the closely related class of total graphs. The total graph of $G$, denoted by $T(G)$, is defined on the vertex set $V(G)\sqcup E(G)$ with $c_1,c_2\in V(G)\sqcup E(G)$ adjacent whenever $c_1$ and $c_2$ are adjacent to or incident on each other in $G$. We first show that there exists a constant $C$ such that, if the connectivity of $G$ is at least $C$, then Hadwiger's conjecture is t…

Theory on Structure and Coloring of Maximal Planar Graphs (I): Relationship between Structure and Coloring

Maximal planar graph refers to the planar graph with the most edges, which means no more edges can be added so that the resulting graph is still planar. The Four-Color Conjecture says that every planar graph without loops is 4-colorable. Indeed, in order to prove Four-Color Conjecture, it clearly suffices to show that all maximal planar graphs are 4-colorable. Since this conjecture was proposed in 1852, no mathematical proofs have been invented up until now. Maybe the main reasons lie in the following three aspects in terms of maximal planar graphs: not clearing up the structures, not figuring…

On the Erd{\H o}s--Hajnal problem in the case of 3-graphs

Let $m(n,r)$ denote the minimal number of edges in an $n$-uniform hypergraph which is not $r$-colorable. For the broad history of the problem see [RaiSh]. It is known that for a fixed $n$ the sequence \[ \frac{m(n,r)}{r^n} \] has a limit. The only trivial case is $n=2$ in which $m(2,r) = \binom{r+1}{2}$. In this note we focus on the case $n=3$. First, we compare the existing methods in this case and then improve the lower bound.

Clustered Variants of Hajós' Conjecture

Hajós conjectured that every graph containing no subdivision of the complete graph $K_{s+1}$ is properly $s$-colorable. This conjecture was disproved by Catlin. Indeed, the maximum chromatic number of such graphs is $Ω(s^2/\log s)$. We prove that $O(s)$ colors are enough for a weakening of this conjecture that only requires every monochromatic component to have bounded size (so-called clustered coloring). Our approach leads to more results. Say that a graph is an almost $(\leq 1)$-subdivision of a graph $H$ if it can be obtained from $H$ by subdividing edges, where at most one edge is subdivid…

Notes on Hamiltonian threshold and chain graphs

We revisit results obtained in [F. Harary, U. Peled, Hamiltonian threshold graphs, Discrete Appl.~Math., 16 (1987), 11--15], where several necessary and necessary and sufficient conditions for a connected threshold graph to be Hamiltonian were obtained. We present these results in new forms, now stated in terms of structural parameters that uniquely define the threshold graph and we extend them to chain graphs. We also identify the chain graph with minimum number of Hamilton cycles within the class of Hamiltonian chain graphs of a given order.

DP-3-coloring of planar graphs without $4,9$-cycles and two cycles from $\{5,6,7,8\}$

A generalization of list-coloring, now known as DP-coloring, was recently introduced by Dvořák and Postle. Essentially, DP-coloring assigns an arbitrary matching between lists of colors at adjacent vertices, as opposed to only matching identical colors as is done for list-coloring. Several results on list-coloring of planar graphs have since been extended to the setting of DP-coloring. We note that list-coloring results do not always extend to DP-coloring results. Our main result in this paper is to prove that every planar graph without cycles of length $\{4, a, b, 9\}$ for $a, b \in \{6, 7, 8…

曲面上的曲线着色

Coloring curves on surfaces

本研究探讨曲面曲线图色数，得出分离曲线色数增长规律、固定同调类曲线唯一着色性，给出曲线色数最佳界，还研究弧图变体及低复杂度曲面精确结果，关联Kneser图等领域。

关于hereditary超图族多色着色的注记II

Note on polychromatic coloring of hereditary hypergraph families II

本文通过构造特定超图，展示了对于h≥3，存在统一性为2h-1的超图，其无多色3-着色但所有h-重限制子超图可2-着色。

关于稀疏图的奇染色

On odd colorings of sparse graphs

本文刻画了最大平均度达到临界值的非无冲突可染色图，改进了奇4-染色和奇6-染色的充分条件，解决了Cranston猜想相关的极值问题。

New results on word-representable graphs

A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $(x,y)\in E$ for each $x\neq y$. The set of word-representable graphs generalizes several important and well-studied graph families, such as circle graphs, comparability graphs, 3-colorable graphs, graphs of vertex degree at most 3, etc. By answering an open question from [M. Halldorsson, S. Kitaev and A. Pyatkin, Alternation graphs, Lect. Notes Comput. Sci. 6986 (2011) 191--202. Proceedings of the 37th International Workshop on Graph-Theoretic …

A note on the conflict-free chromatic index

Let $G$ be a graph with maximum degree $Δ$ and without isolated vertices. An edge colouring $c$ of $G$ is conflict-free if the closed neighbourhood of every edge includes a uniquely coloured element. The least number of colours admitting such $c$ is the conflict-free chromatic index of $G$, denoted by $χ'_{CF}(G)$. In "Conflict-free chromatic number versus conflict-free chromatic index" [J. Graph Theory, 2022; 99: 349--358] it was recently proved by means of the probabilistic method that $χ'_{CF}(G)\leq C_1\log_2Δ+C_2$, where $C_1>337$ and $C_2$ are constants, whereas there are families of gra…

A note on the metric and edge metric dimensions of 2-connected graphs

For a given graph $G$, the metric and edge metric dimensions of $G$, $\dim(G)$ and ${\rm edim}(G)$, are the cardinalities of the smallest possible subsets of vertices in $V(G)$ such that they uniquely identify the vertices and the edges of $G$, respectively, by means of distances. It is already known that metric and edge metric dimensions are not in general comparable. Infinite families of graphs with pendant vertices in which the edge metric dimension is smaller than the metric dimension are already known. In this article, we construct a 2-connected graph $G$ such that $\dim(G)=a$ and ${\rm e…

Some remarks on even-hole-free graphs

A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph has a bisimplicial vertex. Both Hadwiger's conjecture and the Erdős-Lovász Tihany conjecture have been shown to be true for quasi-line graphs, but are open for even-hole-free graphs. In this note, we prove that for all $k\ge7$, every even-hole-free graph with no $K_k$ minor is $(2k-5)$-colorable; every even-hole-free graph $G$ with $ω(G)<χ(G)=s+t-1$ satisfie…