Search ResultsShowing 1-20 of 195
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arXiv:2506.19100 (Published 2025-06-23)
On Gyárfás' Path-Colour Problem
Comments: 30 pages, 5 figuresCategories: math.COIn their 1997 paper titled ``Fruit Salad", Gy\'{a}rf\'{a}s posed the following conjecture: there exists a constant $k$ such that if each path of a graph spans a $3$-colourable subgraph, then the graph is $k$-colourable. It is noted that $k=4$ might suffice. Let $r(G)$ be the maximum chromatic number of any subgraph $H$ of $G$ where $H$ is spanned by a path. The only progress on this conjecture comes from Randerath and Schiermeyer in 2002, who proved that if $G$ is an $n$ vertex graph, then $\chi(G) \leq r(G)\log_{\frac{8}{7}}(n)$. We prove that for all natural numbers $r$, there exists a graph $G$ with $r(G)\leq r$ and $\chi(G)\geq \lfloor\frac{3r}{2}\rfloor -1$. Hence, for all constants $k$ there exists a graph with $\chi - r > k$. Our proof is constructive. We also study this problem in graphs with a forbidden induced subgraph. We show that if $G$ is $K_{1,t}$-free, for $t\geq 4$, then $\chi(G) \leq (t-1)(r(G)+\binom{t-1}{2}-3)$. If $G$ is claw-free, then we prove $\chi(G) \leq 2r(G)$. Additionally, the graphs $G$ where every induced subgraph $G'$ of $G$ satisfy $\chi(G') = r(G')$ are considered. We call such graphs path-perfect, as this class generalizes perfect graphs. We prove that if $H$ is a forest with at most $4$ vertices other than the claw, then every $H$-free graph $G$ has $\chi(G) \leq r(G)+1$. We also prove that if $H$ is additionally not isomorphic to $2K_2$ or $K_2+2K_1$, then all $H$-free graphs are path-perfect.
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Vertex-Based Localization of Turán's Theorem
Let $G$ be a simple graph with $n$ vertices and $m$ edges. According to Tur\'{a}n's theorem, if $G$ is $K_{r+1}$-free, then $m \leq |E(T(n, r))|,$ where $T(n, r)$ denotes the Tur\'{a}n graph on $n$ vertices with a maximum clique of order $r$. A limitation of this statement is that it does not give an expression in terms of $n$ and $r$. A widely used version of Tur\'{a}n's theorem states that for an $n$-vertex $K_{r+1}$-free graph, $m \leq \left\lfloor \frac{n^2(r-1)}{2r} \right\rfloor.$ Though this bound is often more convenient, it is not the same as the original statement. In particular, the class of extremal graphs for this bound, say $\mathcal{S}$, is a proper subset of the set of Tur\'{a}n graphs. In this paper, we generalize this result as follows: For each $v \in V(G)$, let $c(v)$ be the order of the largest clique that contains $v$. We show that \[ m \leq \left\lfloor\frac{n}{2}\sum_{v\in V(G)}\frac{c(v)-1}{c(v)}\right\rfloor\] Furthermore, we characterize the class of extremal graphs that attain equality in this bound. Interestingly, this class contains two extra non-Tur\'{a}n graphs other than the graphs in $\mathcal{S}$.
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Vertex-Based Localization of Erdős-Gallai Theorems for Paths and Cycles
For a simple graph $G$, let $n$ and $m$ denote the number of vertices and edges in $G$, respectively. The Erd\H{o}s-Gallai theorem for paths states that in a simple $P_k$-free graph, $m \leq \frac{n(k-1)}{2}$, where $P_k$ denotes a path with length $k$ (that is, with $k$ edges). In this paper, we generalize this result as follows: For each $v \in V(G)$, let $p(v)$ be the length of the longest path that contains $v$. We show that \[m \leq \sum_{v \in V(G)} \frac{p(v)}{2}\] The Erd\H{o}s-Gallai theorem for cycles states that in a simple graph $G$ with circumference (that is, the length of the longest cycle) at most $k$, we have $m \leq \frac{k(n-1)}{2}$. We strengthen this result as follows: For each $v \in V(G)$, let $c(v)$ be the length of the longest cycle that contains $v$, or $2$ if $v$ is not part of any cycle. We prove that \[m \leq \left( \sum_{v \in V(G)} \frac{c(v)}{2} \right) - \frac{c(u)}{2}\] where $c(u)$ denotes the circumference of $G$. \newline Furthermore, we characterize the class of extremal graphs that attain equality in these bounds.
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arXiv:2502.09576 (Published 2025-02-13)
Interpolating chromatic and homomorphism thresholds
Comments: 29 pagesCategories: math.COThe problem of chromatic thresholds seeks for minimum degree conditions that ensure $H$-free graphs to have a bounded chromatic number, or equivalently a bounded size homomorphic image. The strengthened homomorphism thresholds problem further requires that the homomorphic image itself is $H$-free. The purpose of this paper is two-fold. First, we define a generalized notion of threshold which encapsulates and interpolates chromatic and homomorphism thresholds via the theory of VC-dimension. Our first result shows a smooth transition between these two thresholds when varying the restrictions on homomorphic images. In particular, we proved that for $t \ge s \ge 3$ and $\epsilon>0$, if $G$ is an $n$-vertex $K_s$-free graph with VC-dimension $d$ and $\delta(G) \ge (\frac{(s-3)(t-s+2)+1}{(s-2)(t-s+2)+1} + \epsilon)n$, then $G$ is homomorphic to a $K_t$-free graph $H$ with $|H| = O(1)$. Moreover, we construct graphs showing that this minimum degree condition is optimal. This extends and unifies the results of Thomassen, {\L}uczak and Thomass\'e, and Goddard, Lyle and Nikiforov, and provides a deeper insight into the cause of existences of homomorphic images with various properties. Second, we introduce the blowup threshold $\delta_B(H)$ as the infimum $\alpha$ such that every $n$-vertex maximal $H$-free graph $G$ with $\delta(G)\ge\alpha n$ is a blowup of some $F$ with $|F|=O(1)$. This notion strengthens homomorphism threshold. While the homomorphism thresholds for odd cycles remain unknown, we prove that $\delta_B(C_{2k-1})=1/(2k-1)$ for any integer $k\ge 2$. This strengthens the result of Ebsen and Schacht and answers a question of Schacht and shows that, in sharp contrast to the chromatic thresholds, 0 is an accumulation point for blowup thresholds. Our proofs mix tools from VC-dimension theory and an iterative refining process, and draw connection to a problem concerning codes on graphs.
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arXiv:2501.13907 (Published 2025-01-23)
Graphs with no long claws: An improved bound for the analog of the Gyárfás' path argument
Subjects: G.2.2For a fixed integer $t \geq 1$, a ($t$-)long claw, denoted $S_{t,t,t}$, is the unique tree with three leaves, each at distance exactly $t$ from the vertex of degree three. Majewski et al. [ICALP 2022, ACM ToCT 2024] proved an analog of the Gy\'{a}rf\'{a}s' path argument for $S_{t,t,t}$-free graphs: given an $n$-vertex $S_{t,t,t}$-free graph, one can delete neighborhoods of $\mathcal{O}(\log n)$ vertices so that the remainder admits an extended strip decomposition (an appropriate generalization of partition into connected components) into particles of multiplicatively smaller size. This statement has proven to be very useful in designing quasi-polynomial time algorithms for Maximum Weight Independent Set and related problems in $S_{t,t,t}$-free graphs. In this work, we refine the argument of Majewski et al. and show that a constant number of neighborhoods suffice.
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arXiv:2501.13036 (Published 2025-01-22)
Even cycles in graphs avoiding longer even cycles
Comments: 10 pagesCategories: math.COA conjecture of Verstra\"ete states that for any fixed $\ell < k$ there exists a positive constant $c$ such that any $C_{2k}$-free graph $G$ contains a $C_{2\ell}$-free subgraph with at least $c |E(G)|$ edges. For $\ell = 2$, this conjecture was verified by K\"uhn and Osthus. We show that $C_6$ and $C_{2k}$ satisfy the conjecture for all odd $k$, but observe that a recent construction of a dense $C_{10}$-free subgraph of the hypercube yields a counterexample to the conjecture for $C_8$ and $C_{10}$.
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arXiv:2501.02543 (Published 2025-01-05)
Coloring of ($P_2+P_4$, $K_4-e$)-free graphs
For a graph $G$, $\chi(G)$ and $\omega(G)$ respectively denote the chromatic number and clique number of $G$. In this paper, we show that if $G$ is a ($P_2+P_4$, $K_4-e$)-free graph with $\omega(G)\geq 3$, then $\chi(G)\leq \max\{6, \omega(G)\}$, and that the bound is tight for $\omega(G)\notin \{4,5\}$. These extend the results known for the class of ($P_2+P_3$, $K_4-e$)-free graphs, improves the bound of Chen and Zhang [arXiv:2412.14524[math.CO], 2024] given for the class of ($P_2+P_4$, $K_4-e$)-free graphs, partially answers a question of Ju and Huang [Theor. Comp. Sci. 993 (2024) Article No.: 114465] on `near optimal colorable graphs', and partially answers a question of Schiermeyer (unpublished) on the chromatic bound for ($P_7$, $K_4-e$)-free graphs.
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arXiv:2412.13792 (Published 2024-12-18)
Spectral radius of graphs of given size with forbidden a fan graph $F_6$
Comments: 21 pagesCategories: math.COLet $F_k=K_1\vee P_{k-1}$ be the fan graph on $k$ vertices. A graph is said to be $F_k$-free if it does not contain $F_k$ as a subgraph. Yu et al. in [arXiv:2404.03423] conjectured that for $k\geq2$ and $m$ sufficiently large, if $G$ is an $F_{2k+1}$-free or $F_{2k+2}$-free graph, then $\lambda(G)\leq \frac{k-1+\sqrt{4m-k^2+1}}{2}$ and the equality holds if and only if $G\cong K_k\vee\left(\frac{m}{k}-\frac{k-1}{2}\right)K_1$. Recently, Li et al. in [arXiv:2409.15918] showed that the above conjecture holds for $k\geq 3$. The only left case is for $k=2$, which corresponds to $F_5$ or $F_6$. Since the case of $F_5$ was solved by Yu et al. in [arXiv:2404.03423] and Zhang and Wang in [On the spectral radius of graphs without a gem, Discrete Math. 347 (2024) 114171]. So, one needs only to deal with the case of $F_6$. In this paper, we solve the only left case by determining the maximum spectral radius of $F_6$-free graphs with size $m\geq 88$, and the corresponding extremal graph.
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arXiv:2411.14364 (Published 2024-11-21)
A new lower bound for the multicolor Ramsey number $r_k(K_{2, t + 1})$
Comments: 8 pagesCategories: math.COIn this short note, we provide a new infinite family of $K_{2, t+1}$-free graphs for each prime power $t$. Using these graphs, we show that it is possible to partition the edges of $K_n$ into parts, such that each part is isomorphic to our $K_{2, t+1}$-free graph. This yields an improved lower bound to the multicolor Ramsey number $r_k(K_{2, t+1})$ when $k$ and $t$ are powers of the same prime. For these values of $k$ and $t$, our coloring implies that $$ tk^2 + 1 \leq r_k(K_{2, t+1}) \leq tk^2 + k + 2. $$ where the upper bound is due to Chung and Graham.
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arXiv:2410.18309 (Published 2024-10-23)
Every $3$-connected $\{K_{1,3},Γ_3\}$-free graph is Hamilton-connected
Categories: math.COWe show that every $3$-connected $\{K_{1,3},\Gamma_3\}$-free graph is Hamilton-connected, where $\Gamma_3$ is the graph obtained by joining two vertex-disjoint triangles with a path of length $3$. This resolves one of the two last open cases in the characterization of pairs of connected forbidden subgraphs implying Hamilton-connectedness. The proof is based on a new closure technique, developed in a previous paper, and on a structural analysis of small subgraphs, cycles and paths in line graphs of multigraphs. The most technical steps of the analysis are computer-assisted. Keywords: Hamilton-connected; closure; forbidden subgraph; claw-free; $\Gamma_3$-free
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arXiv:2410.07721 (Published 2024-10-10)
The maximum spectral radius of $θ_{1,3,3}$-free graphs with given size
Comments: 14 pagesCategories: math.COA graph $G$ is said to be $F$-free if it does not contain $F$ as a subgraph. A theta graph, say $\theta_{l_1,l_2,l_3}$, is the graph obtained by connecting two distinct vertices with three internally disjoint paths of length $l_1, l_2, l_3$, where $l_1\leq l_2\leq l_3$ and $l_2\geq2$. Recently, Li, Zhao and Zou [arXiv:2409.15918v1] characterized the $\theta_{1,p,q}$-free graph of size $m$ having the largest spectral radius, where $q\geq p\geq3$ and $p+q\geq2k+1\geq7$, and proposed a problem on characterizing the graphs with the maximum spectral radius among $\theta_{1,3,3}$-free graphs. In this paper, we consider this problem and determine the maximum spectral radius of $\theta_{1,3,3}$-free graphs with size $m$ and characterize the extremal graph. Up to now, all the graphs in $\mathcal{G}(m,\theta_{1,p,q})$ which have the largest spectral radius have been determined, where $q\geq p\geq 2$.
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arXiv:2409.14138 (Published 2024-09-21)
A Brualdi-Hoffman-Turán problem for friendship graph
Comments: This article is a draft version and may have some clerical and grammatical errorsCategories: math.COA graph is said to be $H$-free if it does not contain $H$ as a subgraph. Brualdi-Hoffman-Tur\'{a}n type problem is to determine the maximum spectral radius of an $H$-free graph $G$ with give size $m$. The $F_k$ is the graph consisting of $k$ triangles that intersect in exactly one common vertex, which is known as the friendship graph. In this paper, we resolve a conjecture (the Brualdi-Hoffman-Tur\'{a}n-type problem for $F_k$) of Li, Lu and Peng [Discrete Math. 346 (2023) 113680] by using the $k$-core technique presented in Li, Zhai and Shu [European J. Combin, 120 (2024) 103966].
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arXiv:2409.13161 (Published 2024-09-20)
Frozen colourings in $2K_2$-free graphs
Comments: 18 pagesCategories: math.COThe \emph{reconfiguration graph of the $k$-colourings} of a graph $G$, denoted $\mathcal{R}_k(G)$, is the graph whose vertices are the $k$-colourings of $G$ and two vertices of $\mathcal{R}_k(G)$ are joined by an edge if the colourings of $G$ they correspond to differ in colour on exactly one vertex. A $k$-colouring of a graph $G$ is called \emph{frozen} if it is an isolated vertex in $\mathcal{R}_k(G)$; in other words, for every vertex $v \in V(G)$, $v$ is adjacent to a vertex of every colour different from its colour. A clique partition is a partition of the vertices of a graph into cliques. A clique partition is called a $k$-clique-partition if it contains at most $k$ cliques. Clearly, a $k$-colouring of a graph $G$ corresponds precisely to a $k$-clique-partition of its complement, $\overline{G}$. A $k$-clique-partition $\mathcal{Q}$ of a graph $H$ is called \emph{frozen} if for every vertex $v \in V(H)$, $v$ has a non-neighbour in each of the cliques of $\mathcal{Q}$ other than the one containing $v$. The cycle on four vertices, $C_4$, is sometimes called the \emph{square}; its complement is called $2K_2$. We give several infinite classes of $2K_2$-free graphs with frozen colourings. We give an operation which transforms a $k$-chromatic graph with a frozen $(k+1)$-colouring into a $(k+1)$-chromatic graph with a frozen $(k+2)$-colouring. Our operation preserves being $2K_2$-free. It follows that for all $k \ge 4$, there is a $k$-chromatic $2K_2$-free graph with a frozen $(k+1)$-colouring. We prove these results by studying frozen clique partitions in $C_4$-free graphs. We say a graph $G$ is \emph{recolourable} if $R_{\ell}(G)$ is connected for all $\ell$ greater than the chromatic number of $G$. We prove that every 3-chromatic $2K_2$-free graph is recolourable.
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arXiv:2409.10129 (Published 2024-09-16)
Generalized Turán problem for a path and a clique
Categories: math.COLet $\mathcal{H}$ be a family of graphs. The generalized Tur\'an number $ex(n, K_r, \mathcal{H})$ is the maximum number of copies of the clique $K_r$ in any $n$-vertex $\mathcal{H}$-free graph. In this paper, we determine the value of $ex(n, K_r, \{P_k, K_m \} )$ for sufficiently large $n$ with an exceptional case, and characterize all corresponding extremal graphs, which generalizes and strengthens the results of Katona and Xiao [EJC, 2024] on $ex(n, K_2, \{P_k, K_m \} )$. For the exceptional case, we obtain a tight upper bound for $ex(n, K_r, \{P_k, K_m \} )$ that confirms a conjecture on $ex(n, K_2, \{P_k, K_m \} )$ posed by Katona and Xiao.
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arXiv:2409.06650 (Published 2024-09-10)
Induced subgraphs of $K_r$-free graphs and the Erdős--Rogers problem
Categories: math.COFor two graphs $F,H$ and a positive integer $n$, the function $f_{F,H}(n)$ denotes the largest $m$ such that every $H$-free graph on $n$ vertices contains an $F$-free induced subgraph on $m$ vertices. This function has been extensively studied in the last 60 years when $F$ and $H$ are cliques and became known as the Erd\H{o}s-Rogers function. Recently, Balogh, Chen and Luo, and Mubayi and Verstra\"ete initiated the systematic study of this function in the case where $F$ is a general graph. Answering, in a strong form, a question of Mubayi and Verstra\"ete, we prove that for every positive integer $r$ and every $K_{r-1}$-free graph $F$, there exists some $\varepsilon_F>0$ such that $f_{F,K_r}(n)=O(n^{1/2-\varepsilon_F})$. This result is tight in two ways. Firstly, it is no longer true if $F$ contains $K_{r-1}$ as a subgraph. Secondly, we show that for all $r\geq 4$ and $\varepsilon>0$, there exists a $K_{r-1}$-free graph $F$ for which $f_{F,K_r}(n)=\Omega(n^{1/2-\varepsilon})$. Along the way of proving this, we show in particular that for every graph $F$ with minimum degree $t$, we have $f_{F,K_4}(n)=\Omega(n^{1/2-6/\sqrt{t}})$. This answers (in a strong form) another question of Mubayi and Verstra\"ete. Finally, we prove that there exist absolute constants $0<c<C$ such that for each $r\geq 4$, if $F$ is a bipartite graph with sufficiently large minimum degree, then $\Omega(n^{\frac{c}{\log r}})\leq f_{F,K_r}(n)\leq O(n^{\frac{C}{\log r}})$. This shows that for graphs $F$ with large minimum degree, the behaviour of $f_{F,K_r}(n)$ is drastically different from that of the corresponding off-diagonal Ramsey number $f_{K_2,K_r}(n)$.
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arXiv:2409.04042 (Published 2024-09-06)
On a conjecture of Kim, Kim and Liu
Comments: 32 pagesCategories: math.COIn 1969, Erd\H{o}s and S\'{o}s initiated the study of the Ramsey-Tur\'{a}n type problems: Determine the maximum number of edges of an $n$-vertex $K_{p+1}$-free graph without large independent set. Given integers $p, q\ge2$, we say that a graph $G$ is $(K_p,K_q)$-free if there exists a red/blue edge coloring of $G$ such that it contains neither a red $K_p$ nor a blue $K_q$. Fix a function $f( n )$, the Ramsey-Tur\'{a}n number $RT( {n,p,q,f( n ))} $ is the maximum number of edges in an $n$-vertex $(K_p,K_q)$-free graph with independence number at most $f( n )$. For any $\delta>0$, let $\rho (p, q,\delta ) = \mathop {\lim }\limits_{n \to \infty } \frac{RT(n,p, q,\delta n)}{n^2}$. Kim, Kim and Liu (2019) determined the value of $\rho(3,p,\delta)$ for $p=3,4,5$ and sufficiently small $\delta>0$. Moreover, they showed $\rho(3,6,\delta)\ge \frac{5}{12}+\frac{\delta}{2}+2\delta^2$ from a skilful construction and conjectured the equality holds for sufficiently small $\delta>0$. In this paper, we obtain $\rho(3,6,\delta)\le\frac{5}{{12}} + \frac{\delta }{2}+ 2.1025\delta ^2$ for sufficiently small $\delta>0$, which is pretty close to solving the conjecture. The key step of the proof is to establish a stability lemma of the $n$-vertex graph that is $(K_3,K_6)$-free with independence number $\delta n$.
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arXiv:2408.15487 (Published 2024-08-28)
A strong structural stability of $C_{2k+1}$-free graphs
Categories: math.COF\"uredi and Gunderson showed that $ex(n, C_{2k+1})$ is achieved only on $K_{\lfloor\frac{n}{2}\rfloor, \lceil\frac{n}{2}\rceil}$ if $n\ge 4k-2$. It is natural to study how far a $ C_{2k+1}$-free graph is from being bipartite.Let $T^*(r, n)$ be obtained by adding a suspension $K_{r}$ with $1$ suspension point to $K_{\lfloor\frac{n-r+1}{2}\rfloor, \lceil\frac{n-r+1}{2}\rceil}$. We show that for integers $r, k$ with $3\le r\le 2k-4$ and $n\ge 20(r+2)^2k$, if $G$ is a $C_{2k+1}$-free $n$-vertex graph with $e(G)\ge e(T^*(r, n))$, then $G$ is obtained by adding suspensions to a bipartite graph one by one and the total number of vertices in all suspensions minus intersection points is no more than $r-1$. In other words, $G=B\bigcup\limits_{i=1}^p G_i$, where $B$ is a bipartite graph, $G_1$ is a suspension to $B$, $G_j$ is a suspension to $B\bigcup\limits_{i=1}^{j-1} G_i$ for $2\le j\le p$ and $\sum\limits_{i=1}^p \vert V(G_i)-V(G_i)\cap V(B\bigcup\limits_{i=1}^{j-1} G_i) \vert\le r-1$. Furthermore, $\sum\limits_{i=1}^p \vert V(G_i)-V(G_i)\cap V(B\bigcup\limits_{i=1}^{j-1} G_i) \vert= r-1$ if and only if $G=T^*(r, n)$. Let $d_2(G)=\min\{|T|: T\subseteq V(G), G-T \ \text{is bipartite}\}$ and $\gamma_2(G)=\min\{|E|: E\subseteq E(G), G-E \ \text{is bipartite}\}$. Our structural stability result implies that $d_2(G)\le r-1$ and $\gamma_2(G)\le {\lceil\frac{r}{2}\rceil \choose 2}+{\lfloor\frac{r}{2}\rfloor \choose 2}$ under the same condition, which is a recent result of Ren-Wang-Wang-Yang [SIAM J. Discrete Math. 38 (2024)]. They proved $d_2(G)\le r-1$ and $\gamma_2(G)\le {\lceil\frac{r}{2}\rceil \choose 2}+{\lfloor\frac{r}{2}\rfloor \choose 2}$ separately. We introduce a new concept strong-$2k$-core which is the key that we can give a stronger structural stability result but a simpler proof.
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arXiv:2408.02400 (Published 2024-08-05)
Disproof of a conjecture by Erdős, Gimbel and Straight
Comments: 4 pagesCategories: math.COThe cochromatic number $\zeta(G)$ of a graph $G$ is the smallest number of colors in a vertex-coloring of $G$ such that every color class induces an independent set or a clique. In 1988, Erd\H{o}s, Gimbel and Straight conjectured that every $K_5$-free graph $G$ with $\zeta(G)>3$ satisfies $\chi(G)\le \zeta(G)+2$. In this note, we present a counterexample to this conjecture and discuss related results and questions.
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arXiv:2407.19149 (Published 2024-07-27)
A Fan-type condition for cycles in $1$-tough and $k$-connected $(P_2\cup kP_1)$-free graphs
Comments: 19 pages, 4 figuresCategories: math.COFor a graph $G$, let $\mu_k(G):=\min~\{\max_{x\in S}d_G(x):~S\in \mathcal{S}_k\}$, where $\mathcal{S}_k$ is the set consisting of all independent sets $\{u_1,\ldots,u_k\}$ of $G$ such that some vertex, say $u_i$ ($1\leq i\leq k$), is at distance two from every other vertex in it. A graph $G$ is $1$-tough if for each cut set $S\subseteq V(G)$, $G-S$ has at most $|S|$ components. Recently, Shi and Shan \cite{Shi} conjectured that for each integer $k\geq 4$, being $2k$-connected is sufficient for $1$-tough $(P_2\cup kP_1)$-free graphs to be hamiltonian, which was confirmed by Xu et al. \cite{Xu} and Ota and Sanka \cite{Ota2}, respectively. In this article, we generalize the above results through the following Fan-type theorem: Let $k$ be an integer with $k\geq 2$ and let $G$ be a $1$-tough and $k$-connected $(P_2\cup kP_1)$-free graph with $\mu_{k+1}(G)\geq\frac{7k-6}{5}$, then $G$ is hamiltonian or the Petersen graph.
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arXiv:2407.01625 (Published 2024-06-29)
Balanced clique subdivisions and cycles lengths in $K_{s, t}$-free graphs
Comments: arXiv admin note: text overlap with arXiv:2010.15802 by other authorsCategories: math.COLet $ t\ge s\ge2$ be integers. Confirming a conjecture of Mader, Liu and Montgomery [J. Lond. Math. Soc., 2017] showed that every $K_{s, t}$-free graph with average degree $d$ contains a subdivision of a clique with at least $\Omega(d^{\frac{s}{2(s-1)}})$ vertices. We give an improvement by showing that such a graph contains a balanced subdivision of a clique with the same order, where a balanced subdivision is a subdivision in which each edge is subdivided the same number of times. In 1975, Erd\H{o}s asked whether the sum of the reciprocals of the cycle lengths in a graph with infinite average degree $d$ is necessarily infinite. Recently, Liu and Montgomery [J. Amer. Math. Soc., 2023] confirmed the asymptotically correct lower bound on the reciprocals of the cycle lengths, and provided a lower bound of at least $(\frac{1}{2} -o_d(1)) \log d$. In this paper, we improve this low bound to $\left(\frac{s}{2(s-1)} -o_d(1)\right) \log d$ for $K_{s, t}$-free graphs. Both proofs of our results use the graph sublinear expansion property as well as some novel structural techniques.